Food physics: physical properties - measurement and application

Food Physics Physical Properties – Measurement and Applications

Ludger O. Figura Arthur A.Teixeira

Food Physics Physical Properties – Measurement and Applications With 131 Figures and 208 Tables

Professor Dr. Ludger O. Figura Hochschule Bremerhaven University of Applied Sciences An der Karlstadt 8 27568 Bremerhaven Germany [email protected] Professor Arthur A. Teixeira Ph.D., P.E. Agricultural and Biological Engineering Department University of Florida 207 Frazier Rogers Hall P.O. Box 110570 Gainesville, FL 32611-0570 USA [email protected]

Library of Congress Control Number: 2007925693 ISBN 978-3-540-34191-8 Springer Berlin Heidelberg New York DOI 10.1007/b107120 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permissions for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover-Design: WMXDesign GmbH, Heidelberg Typesetting: PTP-Berlin Protago-TEX-Production GmbH, Germany Production: LE-TEX Jelonek, Schmidt & Vockler ¨ GbR, Leipzig, Germany Printed on acid-free paper 52/3180/YL - 5 4 3 2 1 0

Foreword

This new book, Food Physics: Physical Properties – Measurement and Applications, is an expanded version of the text in German, Lebensmittelphysik (Springer, 2004), by the first author, Ludger Figura. It is the result of collaboration between Ludger Figura and Art Teixeira who have been teaching food physics and physical properties of foods at the Bremerhaven Hochschule (Germany) and the University of Florida (USA), respectively. The book is a timely addition that will serve as a useful resource on the physics and physical properties of foods. It should be useful worldwide to teach junior- or senior-level undergraduate students. In addition, it should find use in food companies because, as the authors point out: “it is essential that food companies be able to design and control their processing operations to assure maximum product quality and safety and to develop new and improved food products and quality attributes desired by the consuming public.” There are fourteen chapters, in order: Water Activity, Mass and Density, Geometric Properties: Size and Shape, Rheological Properties, Interfacial Phenomena, Permeability, Thermal Properties, including Heat Transfer in Food, Electrical Properties, Magnetic Properties, Electromagnetic Properties, Optical Properties,Acoustical Properties, Radioactivity, and On-Line Sensing. Each subject was given its due weight. The first seven chapters cover about 62% of the book. In addition, there are several appendices on relevant topics, such as: Units and their Conversion, Distribution Functions, Complex Numbers, Greek Letters, Properties of Water, and Conversion Charts for: Temperature, Sugar Concentration, as well as Relevant Literature references. I enjoyed reading an early draft of Food Physics: Physical Properties – Measurement and Applications. I am sure that students and researchers of Food Physics and Physical Properties will find it to be a useful and worthy text. M.A. Rao Emeritus Professor (Active), Food Engineering Cornell University, Geneva, NY

Preface

Why should there be a book about food physics and the physical properties of foods? In order for the food processing industry to increase food safety and to be competitive in an ever expanding global market place, it is essential that food companies be able to design and control their processing operations to assure maximum product quality and safety and to develop new and improved food products with quality attributes desired by the consuming public. The food scientists and engineers entrusted with the responsibility for developing the means by which these results can be achieved will have to have mastered a fundamental knowledge base in the physical properties of food materials, and the science of food physics,which provides the scientific principles upon which these properties can be understood and applied. This book was conceived with this purpose in mind. The book is intended for both food scientists and food engineers, as reflected in the chosen title and subtitle for the book. The title Food Physics is directed to food scientists who recognize the importance of food physics as a core part of a food science curriculum, along side food chemistry and food microbiology, for understanding the physical behavior of food materials. The subtitle Physical Properties – Measurement and Applications is directed to food engineers who are always in need of physical properties for process design and control applications,and recognize that such physical properties can only come from the study of food physics. In fact, it has been the relatively recent introduction of food engineering into the food process industries over the past few decades that called attention to the need for physical properties and the study of food physics in the combined fields of food science and engineering. Food physics offers considerable breadth in the range of topics covered, as well as depth of coverage in each topic. The book contains fourteen chapters with each chapter related to a different field of physics in which physical properties of foods are important. Nearly all areas of physics are covered, beginning with water activity and the role of moisture content in foods, followed by basic properties of mass and density and size and shape, and then continuing through mechanical,rheological,thermal,and electromagnetic radiation properties and their applications, including electrical, magnetic, optical, acoustical, and ionizing radiation properties. The final chapter of the book introduces the reader to the exciting new world of in-line sensors for the on-line measurement of physical phenomena that can be used as indirect indicators of food

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properties or quality attributes that must be controlled in closed-loop feed back control systems for on-line process control in food process automation. The material in each chapter is presented at several levels of depth so that the book can serve as an instructional text for students at one level, a source book on theory and scientific principles for researchers at another level, and a handy reference book for practicing professionals in the field on a third level. The presentation of material in each chapter has been crafted with the undergraduate college student in mind first. Basic scientific principles and theory are explained in simple clear language, drawing on examples of every day life experiences to help students understand the concepts. The derivation of mathematical expressions is carried out in a step-by-step sequence of logic so that students can fully appreciate the subsequent use of these expressions in making the necessary calculations, and most chapters include examples for the students to gain exercise in the calculations. Each chapter also contains further discussion of scientific principles and theory with suggestions and examples of possible new applications with the research graduate student and scientist in mind. Also included in each chapter are cited references to which the reader may go for more detailed information on specific applications of the related physical properties. The authors have combined their experience of more than thirty years teaching food properties to undergraduate food science and engineering students to make this book possible. The book, itself, is primarily an English translation of the recent German text Lebensmittelphysik by Figura, published by Springer in 2004. At the time that first book was published, Figura and Teixeira had already teamed up to begin collaboration on an English language version of the book through a series of exchange visits. This new book, Food Physics: Physical Properties – Measurement and Applications is the result of that collaboration. Quakenbr¨uck, Germany June 2007

Ludger Figura Arthur Teixeira

Contents

1 Water Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Time to Reach Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Solid–Fluid Boundary Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Adsorption Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Surface Adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Sorption Isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Freundlich Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Langmuir Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 BET Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Sorption of Water Vapor in Foods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.7 Water Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.8 Moisture Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.9 Hygroscopicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.10 BET Equation for Foods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.11 GAB Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.12 Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Shelf Life of Food Related to Water Activity . . . . . . . . . . . . . . . . . . . 1.4 Laboratory Determination of Sorption Isotherms . . . . . . . . . . . . . 1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 3 4 5 9 9 10 11 15 17 17 19 20 26 29 30 33 38 38

2 Mass and Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Weighing and Atmospheric Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Temperature Dependency of Density . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Pressure Dependency of Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Specific Gravity (Relative Density) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Methods for Laboratory Measurement of Density . . . . . . . . . . . . . 2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 42 45 46 48 50 51 71 71

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3 Geometric Properties: Size and Shape . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Particle Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Sizing by Image Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Equivalent Diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Geometric Equivalent Diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Physical Equivalent Diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Specific Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5.1 Specific Surface of Individual Particles . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5.2 Specific Surface Area in Bulk Materials . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Particle Shape and Size for Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6.1 Form Factor – Sphericity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Particle Size Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Sizing by Sieving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Modal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Average Particle Size – Integral Mean . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Specific Surface Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Sauter Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.7 Characteristics of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Measuring Particle Size by Other Techniques . . . . . . . . . . . . . . . . . . 3.3.1 Weighing Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Sedimentation and Aerodynamic Classification with Fluids . . . 3.3.3 Optical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Electrical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Other Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 75 77 78 79 80 80 80 81 83 84 87 89 95 96 97 103 104 105 107 107 108 110 111 112 114 114

4 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.2 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5

117 117 118 121 124 126 127 129 132 133 137 139 140 141

Rheological Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniaxial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Young’s Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bulk Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shear Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poisson’s Ratio and Transverse Strain . . . . . . . . . . . . . . . . . . . . . . . . . Rheological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscous Behavior – Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shear Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Newtonian Flow Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-Newtonian Flow Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Newtonian with Non-Newtonian Fluids . . . . . . . Pseudoplastic Flow Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.3.6 Thixotropic Flow Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.7 Dilatant Flow Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.8 Rheopectic Flow Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.9 Plastic Flow Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.10 Overview: Non-Newtonian Flow Behavior . . . . . . . . . . . . . . . . . . . . . 4.3.11 Model Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.12 Ostwald–de-Waele Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.13 Model Functions for Plastic Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Temperature Dependency of Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Measurement of Rheological Properties . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Rotational Rheometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Measuring Instruments Based on Other Principles . . . . . . . . . . . . 4.5.3 Funnel Flow from Beaker or Cup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Stress Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Oscillation Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Rheology and Texture of Solid Foods . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Rheological Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Texture Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

142 142 143 143 145 146 148 151 153 155 155 168 171 173 176 181 185 189 189 196 203 203

5 Interfacial Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Interfacial Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Curved (Convex / Concave) Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Temperature Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Concentration Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Liquid–Liquid–Gas Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5 Solid–Liquid–Gas systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.6 Kinetics of Interfacial Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.7 Adsorption Kinetics at Solid Interfaces . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Measuring Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Measuring Contact Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Dynamic Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207 208 210 213 217 219 221 222 223 223 223 229 229 230 231

6 6.1 6.2 6.3

233 233 237 238

Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady State Diffusion in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conductivity, Conductance and Resistance . . . . . . . . . . . . . . . . . . . . Transport Through Solid Multilayers . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.4 Food Packaging Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Molecular Transport in Permeation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Temperature Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Measurement of Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Analogous Transport Phenomena (Heat and Electricity) . . . . . . 6.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241 245 246 247 251 252 254

7 7.1 7.2 7.3 7.3.1 7.4 7.4.1 7.4.2 7.5 7.6 7.6.1 7.6.2 7.6.3 7.6.4 7.6.5 7.6.6 7.7 7.7.1

Thermal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat and Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamics – Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . Laws of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ideal Gases and Ideal Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Capacity of Real Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Transfer in Food . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conduction Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convection Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Transfer by Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of Thermal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of Thermal Conductivity and Thermal Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Caloric Value of Foods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.1 Caloric (Energy) Requirement of the Human Body . . . . . . . . . . . . 7.8.2 Caloric Value of Food . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.3 Measurement of Caloric (Combustion) Values . . . . . . . . . . . . . . . . . 7.9 Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.1 Thermogravimetry (TG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.2 Heat Flow Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

257 259 260 263 263 265 267 269 270 274 275 276 286 289 290 297 298

8 8.1 8.1.1 8.1.2 8.1.3 8.1.4

333 333 335 336 337 337

Electrical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Dependency of Electrical Conductivity . . . . . . . . . . Solid Foods of Plant Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Foods of Animal Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrolyte Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

298 303 303 306 307 308 308 314 326 327

Contents

XIII

8.2 Measurement of Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . 8.3 Capacitance and Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

343 346 349 350

9 Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Applications for Magnetic Field Forces . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 High-Resolution NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Low-Resolution NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

353 353 353 354 355 356 360 362 367 367 370 371

10 Electromagnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Electric Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Temperature Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Frequency Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Microwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Conversion of Microwaves into Heat . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Penetration Depth of Microwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Microwave Heating of Food . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

373 373 377 378 380 382 383 385 387 387

11 11.1 11.1.1 11.1.2 11.1.3 11.2 11.2.1 11.2.2 11.2.3 11.2.4 11.3 11.4 11.4.1 11.4.2

391 391 391 393 395 395 396 398 399 402 403 404 404 408

Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of Refraction Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications for Refraction Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Colorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Light and Color . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physiology of Color Perception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Color as a Vector Quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Color Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications for Color Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . Near infrared (NIR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measuring Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XIV

Contents

11.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Ultraviolet (UV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.3 Further Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

410 411 411 411 412 412

12 Acoustical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.2 Loudness and Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.3 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Ultrasonic Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

417 417 418 421 422 423 424 425

13 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Types of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Radioactive Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Measurement of Ionizing Radiation (˛-, ˇ-,  -) . . . . . . . . . . . . . . . 13.4 Natural Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Exposure to the Human Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Irradiation of Food and Packaging Material . . . . . . . . . . . . . . . . . . . 13.4.3 Detection of Food Irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

427 427 428 431 433 436 437 440 441 442

14 On-line Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Control Systems – Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Sensor Types and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Some Sensors of Relevance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Weighing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Density Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 Metal Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.4 Flow Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.5 Refraction Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.6 Sensing Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.7 Chemosensors and Biosensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

445 447 451 454 454 455 458 458 462 462 464 466 466

Contents

XV

15 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The International System of Units (SI) . . . . . . . . . . . . . . . . . . . . . . . . Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Greek Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conversion Chart: Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sugar Conversion Chart: Concentration, Density, Refraction . . Fundamental Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Food Material Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Color Test Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

469 469 476 482 488 489 489 493 493 495 510

16

General Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

1 Water Activity

1.1 Introduction Water is an important component of nearly all food materials, and plays a decisive role in dictating the physical properties, quality and microbial, chemical and biochemical degradation of the food material [1]. For most food materials, unless the moisture content is reduced below 50% (wet basis), much of the water content is freely available to behave physically as pure water with properties such as vapor pressure equal to pure water. As moisture content is lowered further, a point will be reached at which the water becomes less active in that it cannot act physically or chemically as pure water. For example, it cannot freeze or act as a solvent or reactant. In this state, it is considered to be bound water. The way in which water is bound to the internal structure of the food, the degree to which it is freely available to act as a solvent, to vaporize or freeze, or the degree to which it is chemically bound and unavailable can all be reflected by an ability to specify the water activity of a food material.The matrix presented in Table 1.1 attempts to illustrate the range of conditions under which water may be bound and the role it can play under each condition [6]. The water activity of a food can be thought of as the equilibrium relative humidity of the food material. When a food sample comes into equilibrium with the atmosphere surrounding it, the water activity in the food sample becomes equal to the relative humidity of the atmosphere surrounding it. Once this equilibrium is reached, the food sample neither gains nor loses moisture over time. A more comprehensive definition of water activity will be given later in this chapter. 1.1.1 Time to Reach Equilibrium The equilibrium relative humidity described above comes about as a result of a two-step process involving transfer of water vapor between the food sample and the surrounding fluid atmosphere. In one step, water vapor must transfer across the boundary surface of the sample in the form of adsorption or desorption, depending upon whether water vapor is entering or exiting the sample.

capillaries

bodies

examples

of solids

wet surfaces

0

heat of binding

in J/g H2 O

free

wet sand

0

nearly free

mechanical

coarse

none

water in

water

wet filter tissue

0–300

nonstoichiometric

physical

capillaries

water in fine

capillary water

dropping from

water mobility

binding type

binding

description

water droplets

adhering

Table 1.1. Types of water binding and related heats of binding [6]

surfaces

water sitting on

adsorbed water

gelatin gel

starch gel,

0–1000

decreased

atmospheric air

solid surfaces in

100–3300

physicochemical

molecules

water between

solute

water as a

water of

compound

water in a

constitution

hydrate

glucose-mono-

crystalline

300–2200

Ca(OH)2

1000–6000

none mobile

stoichiometric

chemical

to a crystal lattice

water belonging

crystal water

2 1 Water Activity

1.1 Introduction

3

The second step involves diffusion,or absorption,of the water vapor into or out of the interior tissue structure of the sample. These two combined processes require considerable time to be accomplished, especially the diffusion process. A classic method for bringing a food sample to a specified water activity is first to weigh it and then place it in a closed chamber maintained at the controlled relative humidity desired to be reached at equilibrium. Depending on whether the sample has an initial water activity above or below the controlled target relative humidity it will either lose or gain moisture over time as it approaches equilibrium. This loss or gain of moisture will be evident as weight loss or gain, which can be monitored over time by periodically weighing the sample until no further change in weight can be detected from one day to the next. Normally, several days may be required to reach equilibrium. Therefore, it is important to understand more fully the transport phenomena of adsorption, desorption and diffusion, discussed in the following subsections. 1.1.2 Solid{Fluid Boundary Surfaces Recall that the first of the transport phenomena mentioned above (adsorption and desorption) occur on the boundary surface at the interface between the solid food material and fluid atmosphere surrounding it (liquid or gas). Such boundary conditions in mass transfer are known as solid–liquid or solid–gas boundary surfaces.When fluid molecules are attracted to the solid surface and begin to adhere to it, the process is called adsorption with respect to the solid. When the same molecules are already present on the solid surface and are escaping from the surface by being attracted to the fluid phase, the process is called desorption with respect to the solid (Figure 1.2). For example, when a fresh moist food sample is being dried to produce a dehydrated product, it is placed into an environment of very low relative humidity.Under that condition, the free water molecules on the surface begin to escape into the relatively dry fluid atmosphere surrounding it in attempt to reach equilibrium relative humidity, and the food sample is undergoing desorption. Alternatively, when a previously dehydrated food sample is being rehydrated by being placed into a relatively humid environment or surrounded by water, the water molecules from the surrounding fluid will be attracted to the dry surface and begin to adhere to it in attempt to reach equilibrium relative humidity, and the food sample is undergoing adsorption. The rate at which these adsorption and desorption processes take place is governed largely by the physical and chemical characteristics of the surface boundary. For example, how readily the fluid molecules can either adhere or escape from the surface can depend upon special features of the surface physical structure, such as roughness, smoothness, and porosity (having a porous structure affecting absorbance, etc.). Besides physical adsorption, there can also be chemical binding that can enhance or interfere with the surface adsorption / desorption process (chemisorption),as shown in Figure 1.2.These

4

1 Water Activity Figure 1.1. Solid surface in adsorption equilibrium with the surrounding atmosphere (schematic). 1: gaseous phase, 2: sorbate, 3: sorbent

Figure 1.2. Types of binding and terms used in sorption

phenomena of adsorption and absorption will be explored more fully in the following section.

1.2 Adsorption Equilibrium The term adsorption equilibrium refers to the steady state condition that is ultimately reached when adsorption and desorption are going on at the same time. This is a dynamic state, in which molecules are leaving the surface in a desorption process, while other molecules are attaching themselves to the surface in an adsorption process. Eventually, a dynamic equilibrium state is reached when the number of molecules leaving the surface and those attaching

1.2 Adsorption Equilibrium

5

themselves to the surface is the same, and the number of molecules resting on the surface remains constant on average. This is adsorption equilibrium. The various factors affecting the rate at which this equilibrium can come about are discussed in the following subsections. 1.2.1 Surface Adhesion Another way to describe the behavior of water in foods is by the mechanisms of molecular adsorption, corresponding to “bound water” and capillary adsorption corresponding to “free water” described earlier. Molecular adsorption occurs under very low water activity when water molecules adhere to specific points in the molecular structure of the cell walls within the solid food material. When the distance between the water molecule and the cell wall becomes small enough, the force of attraction is large enough to draw the water molecule into the micelle network of the cell wall. The force of attraction at such low moisture content is so high that an “adsorption compression” results in a net decrease in volume of the solid–water aggregate. As the moisture content increases, the molecular attraction lessens and there is a volume increase, which is roughly equal to the volume of water added. Because of the initial adsorption compression, however, the total volume of the aggregate remains smaller than that of the sum of the constituents. Normally, it is undesirable to bring water activity of food materials to such low levels that irreversible damage from adsorption compression will occur. The extent and nature of the surface on which adsorption compression can take place are likely the primary factors governing molecular adsorption. Molecular attraction can be due to electronic and van der Waals’ attraction, but it is mostly due to hydrogen bonding in the case of water in foods. Thus, the greater the number of ionic or polar type molecules, the more water is held in the food material in this form. Molecular adsorption is the primary cause of swelling in hygroscopic food materials, such as starches. At still higher moisture contents, where the vapor pressure has not yet reached the saturation point, most of the available attraction sites have been filled with water molecules, and further holding of water molecules is possible only through the formation of “water bridges,” chains of water molecules extending between those molecules which have been directly adsorbed. Nonporous Surfaces In the case of solid food materials with nonporous surfaces when placed in contact with a gaseous phase at a different relative humidity, adsorption or desorption will take place freely at the solid–fluid boundary surface, as described earlier. During adsorption, gaseous molecules will be attracted to the solid surface and begin diffusion to the interior once the surface becomes saturated with gas-phase molecules. In the case of desorption, water molecules

6

1 Water Activity

from within the solid phase will be attracted to the surface, and freely escape into the gaseous phase once they reach the surface. The degree to which the molecules disperse themselves about the surface before escaping to the gaseous phase will depend upon the strength of bonding at the surface. In the case of a nonporous surface with no interference from porosity, this bonding strength is a function of the energy or enthalpy of adsorption or heat of vaporization. Both adsorption and desorption processes take place at the same time, but very different rates, depending on the initial difference in relative humidity of the two phases. At a constant temperature and partial pressure of the gaseous phase, equilibrium will be reached when the results from both processes compensate for themselves, and conditions at the boundary surface remain constant. If partial pressure of the gaseous phase is increased, the equilibrium is disrupted, and adsorption will begin once again until the surface becomes saturated with gaseous molecules forming a complete monolayer of molecules. This is known as monomolecular adhesion of a complete monolayer. If partial pressure is increased even further, then further adsorption from the gaseous phase causes formation of multiple molecular layers. These multiple layers are held by much weaker bonds, and begin to give rise to term “free water” referred to drying technology that can be most easily evaporated. When only lower layers are present approaching only the monolayer, then the binding forces are very strong giving rise to “bound water” that is difficult to remove by evaporation. Table 1.1 shows how these levels of binding depend upon the way the water is held within the material. Porous Surfaces Surfaces with porous structure contain voids that promote transport of water by capillary adsorption. Capillary adsorption occurs when voids in the cellular structure are of the size to hold water in liquid form by forces of surface tension. Normally, these pore sizes would fall in meso- to macro-pore size ranges listed in Table 1.2 [2]. The size of capillaries that will fill with water under different levels of relative humidity can be estimated by the Kelvin equation below: ln

p 2 ·  · Vm = p0 rp · R · T

with Table 1.2. Pore size classes according to IUPAC [2] Term Micro-pores meso-pores macro-pores

pore radius in nm <1 1. . . 25 > 25

(1.1)

1.2 Adsorption Equilibrium

Vm 1 V V = = = R n · Rs · M m · R s  · Rs

7

(1.2)

and so ln

p 2· = p0 rp ·  · Rs · T

where p p0  R Rs M  T rP n V Vm

(1.3)

vapor pressure above curved interface saturation vapor pressure surface tension at capillary wall in N · m−1 universal gas constant in J · K−1 · mol−1 specific gas constant in J·K−1 · kg−1 molar mass of liquid in kg · mol−1 density of the liquid in kg · m−3 temperature in K pore radius in m amount of liquid substance in mol volume in m3 molar volume in m3 · mol−1

Example 1.1. Relative water vapor pressure in cylindrical pores for water vapor adsorption at room temperature: ln

1 2· 2 · 72.25 · 10−3Nm−1 1 p = · · = −3 p0 rp  · Rs · T rp 999 kg · m · 461.9 J · K−1 · kg−1 · 293.15 K

ln

1.0682 mm p = p0 rp

so

This is Kelvin’s equation for calculating the relative vapor pressure of droplets. Droplets have a convex shaped surface whereas liquid surfaces in capillaries have a concave shaped surface (for details refer to Section 5.1.1 and also Table 5.2). In mathematics the difference is the sign of the radius only. That is why we have to put into Kelvin’s equation here negative values of the cylindrical pores, e.g. ln

p 1.0682 mm p = = 0.899 = −0.10682 → p0 −10 nm p0

Based on this equation, the sizes of capillaries for various relative vapor pressures are given in Table 1.3. We can see from Table 1.3 that the vapor pressure of a liquid in a coarse capillary is slightly lower than that of unbound water. For meso-pores and micro-pores, the vapor pressure depression is remarkable.

8

1 Water Activity

Table 1.3. Relative vapor pressures in cylindrical pores p p0

rP in nm

ln

1 10 100 1000

−1.0682 −0.10682 −0.010682 −0.0010682

p/p0 0.344 0.899 0.989 0.998

Bottle or Flask Shaped Pores Reference to cylindrical-shaped pores in the previous section is only an ideal case to help explain the mechanism of capillary adsorption. A way to explain hysteresis of the sorption isotherm is to assume surfaces having pores with the shape of a bottle or flask, whose opening has a considerably smaller radius than the bottom of the pore (see Figure 1.3). Recall that the relative vapor pressure required for capillary adsorption will depend upon the pore radius. In the case of such bottle-shaped pores, the smaller radius at the top of the pore will govern the vapor pressure needed for the process of desorption (drying), in which water molecules must be drawn out from the bottom of the pore. However, in the case of the adsorption process when the material starts out in the dry form and the pores are initially empty of free water, the water molecules adsorb at the larger radius at the bottom of the pore. This will require a greater vapor pressure to reach the same level of moisture content during desorption than was required during adsorption. This difference in vapor pressure that is needed to reach the same level of moisture content depending on the direction of the process (adsorption or desorption) is often given as a possible explanation for the hysteresis observed in most sorption isotherms of food materials (graph IV in Table 1.4). Sorption isotherms will be discussed at some length in the following sections.

Figure 1.3. The capillary radius on adsorption (I) is different from that on desorption (II) when a pore has the shape of a bottle or flask (schematic)

1.2 Adsorption Equilibrium

9

1.2.2 Sorption Isotherms Sorption isotherms are graphical plots of the equilibrium between surface adhesion forces and the partial pressure of the gaseous adsorbent at the boundary surface over a range of partial pressures at a constant temperature. Four classic types of sorption isotherms encountered in scientific studies are shown in Table 1.4 along with names of the mathematical models that are used to characterize each type [4]. Table 1.4. Four main types of sorption isotherms I

II

III

IV

Freundlich

Langmuir

BET / GAB

BET / GAB with pores

1.2.3 Freundlich Model The Freundlich model is given by equation (1.4), and is intended to characterize the isotherm when it shows nearly no saturation behavior (Type I) when adhesion at the boundary surface takes place. Since the Freundlich model is a simple power law equation, taking logarithms of both sides will produce a linear equation (1.4) from which the Freundlich constants (aF and bF ) in the model can be obtained by linear regression of a log–log plot of equation (1.5). The constant bF is taken from the slope of the straight line log–log plot, and the constant aF can then be found by substitution. The Freundlich model is the model of choice when sorption isotherms are to be analyzed in regions of very low partial pressure. However, when regions of higher partial pressure are important, other models like the Langmuir and BET / GAB models described in the following discussion are better suited. m = aF · pbF

(1.4)

lg m = lg aF + bF · lg p

(1.5)

or where m mass of adsorbent in kg aF , bF Freundlich constants (0 < bF < 1) p partial pressure in Pa

10

1 Water Activity

1.2.4 Langmuir Model The Langmuir model focuses on characterizing the saturation behavior of the sorption process. This is the region in the sorption isotherm where the curve tends to flatten out (Type II). This region of the isotherm is often explained by realizing that it normally covers the range of partial pressures over which molecular adhesion at the boundary surface is a saturated monolayer. Under this condition a relatively wide shift in partial pressure may produce relatively little change in molecular adhesion at the surface. The model is based upon the assumption that in adsorption equilibrium the rates of adsorption k and desorption k must produce an equal end result, as reflected in equations (1.6) and (1.7): k · p · (1 − m) = k · m m=

(1.6)

k·p k · p + k

(1.7)

Over the range of the curve: p m = mmax p+b where M mmax k, k b p

(1.8)

mass of adsorbent in kg maximum mass of adsorbent in kg rate constants Langmuir constant in Pa partial pressure in Pa

The Langmuir model starts out with the assumption of a homogeneous monomolecular layer in which the adsorbent is held with maximum adhesion at the surface mmax . The model parameters (b, mmax ) can be determined by writing the equation in the form of 1 b 1 1 + · = m mmax mmax p This is the equation of a straight line in a graph of

(1.9) 1 m

versus 1p . The slope

b mmax

1 and intercept mmax will give the constants. The saturation behavior described by the Langmuir model applies to gas and liquid phase adsorption, particularly in the case of chemisorption (see Figure 1.2) when the monomolecular layer covering the boundary surface cannot be exceeded. If the monolayer were to become covered with additional molecular layers, this would give rise to multilayer adsorption, as in the case of physisorption (see Figure 1.2). In this case the Langmuir model would fail, and the Brunnauer, Emmet and Teller (BET) model, described next, should be used instead. Table 1.5 gives an overview of the applicability of these various models at increasing partial pressure.

1.2 Adsorption Equilibrium

11

Table 1.5. Adsorption models and its range of application General:

m p = mmax p + b

for small p 1 m = b mmax 1+ p p m ≈ mmax b

for middle p 1

m = mmax

1+

b p

for higher p p m = mmax p + b

m = mmax · k · pbF

p m ≈ = const. mmax p

m=k·p

m = aF · pbF

m = mmax

Henry’s law

Freundlich model

Langmuir model

Note that the Langmuir model changes into the Freundlich model as partial pressures decrease from the intermediate range. At very low partial pressures the Freundlich model becomes practically identical with Henry’s law. 1.2.5 BET Model Once multilayer molecular adsorption is reached, further increase in partial pressures will cause the isotherm to depart from the relatively flat region characterized earlier by the Langmuir model, and it will begin to increase dramatically reflecting the weakening bonds of multilayer adhesion.When the isotherm is examined over the full range of partial pressure, it will take on a sigmoid shape (Types III and IV in Table 1.4). The best known and most widely used mathematical representation of the complete adsorption phenomenon in biological materials is given by the BET equation (1.10), after Brunnauer, Emmet and Teller [5], because it mathematically characterizes the entire isotherm over all three regions: p 1 C−1 p = + · V · (ps − p) Va · C Va · C pS

(1.10)

with V·

1 =m 

and the abbreviation p =' pS it is

(1.11)

(1.12)

12

1 Water Activity

p pS

1 C−1 p =  + · p m a · C m a · C pS m· 1− pS

(1.13)

respectively ' 1 C −1 = + ·' m · (1 − ' ) ma · C ma · C where p  ' Va ma C m V pS

(1.14)

partial pressure in Pa density of the adsorbent in kg · m−3 relative partial pressure volume of the monolayer in m3 mass of the monolayer in kg BET constant mass of adsorbent in kg Volume of adsorbent in m3 saturation vapor pressure in Pa

The BET equation can undergo a linear transformation by creating a graphical ' plot of m1 · 1−' against ' , as shown in Figure 1.4. This produces a straight line having an equation in the form y = a + b · ' . The BET constant C can be obtained from the slope and intercept of this straight line using the expression, C = ab + 1, the monolayer mass ma from equation (1.15): 1 1 = (1.15) ma =  b a+b a +1 a The BET model has a strong advantage over the Freundlich or Langmuir models because of its validity over the full range of partial pressures. It has become a very useful tool for estimating the volume or mass of the monolayer on the surface boundary of powders or materials with porous surfaces with meso-pore size. For example if we adsorb nitrogen onto a powder surface, the

Figure 1.4. BET plot of adsorbed mass versus relative vapor pressure

1.2 Adsorption Equilibrium

13

monolayer area can be calculated from the known area of a nitrogen molecule AN2 = 1.62 · 10−19m2 [38]. If water vapor is used as the adsorbent, then the p relative partial pressure pS = ' takes on the meaning of equilibrium relative humidity (or water activity), which will be explained further. When isotherm data points taken from an adsorption process are plotted on the same graph as those taken from a desorption process on the same material at the same temperature, the two isotherms will follow different pathways revealing the hysteresis behavior shown in Type IV isotherms (Table 1.4). Sorption isotherm hysteresis is far from being understood [136]. As explained earlier, the presence of bottle-shaped pores with openings that are much narrower than their interior is a possible cause of this type of hysteresis. Another possible cause is that irreversible changes may occur in the food material during the adsorption/desorption experiment. Both desorption and adsorption isotherms can be characterized well by the BET equation. The desorption isotherm relates to a drying process, in which a fresh food sample with initial high moisture content is exposed to a gaseous surrounding of lower and lower partial pressures. The desorption isotherm relates to a rehydration process,in which an initially dry food sample is exposed to a surrounding fluid of higher and higher partial pressure. Understanding the distinction between both cases is a necessary prerequisite for undertaking the experimental procedures used to obtain the sorption isotherm for a given material. Thermodynamics of Sorption Isotherms Another feature of the BET isotherms is that thermodynamic information can also be obtained from further analysis. This type of information can be very useful in the engineering design of drying processes for food dehydration. One of the assumptions upon which the model is based is that the adsorbate (water) is initially held in a liquid state within the moist food material to be dried, and must first undergo phase transition to the gaseous or vapor phase before being desorbed from the boundary surface. In order to remove a monolayer of water molecules from the solid boundary surface, the energy required to provide sufficient heat (enthalpy) of vaporization hvap must be delivered to the boundary surface. In addition to the enthalpy of vaporization, energy will be needed to break the adhesion forces bonding the monolayer to the surface boundary, discussed earlier. This monolayer-bonding enthalpy hC will depend upon the chemical and physical characteristics affecting the bonding forces between the two substances at the boundary surface monolayer (see Table 1.1), and must be delivered in excess of the enthalpy of vaporization in order to drive the desorption process. Thus, the total energy required is the monolayer-desorption enthalpy, and is simply the sum of these two enthalpies, as expressed in equation (1.16): hs,mono = hvap + hC

(1.16)

14

1 Water Activity

In the reverse direction when the substance is undergoing the adsorption process,water vapor molecules must first condense into liquid at the monolayer surface, giving up their enthalpy of vaporization, as well as the enthalpy for building the monolayers (monolayer-building enthalpy). These enthalpies are given up freely during the adsorption process, and together are known as the specific sorption enthalpy of the monolayers hs,mono. Recall that sorption isotherms are obtained and reported at a constant temperature. When obtained at different constant temperatures, they will become different for the same substance. The constant C in the BET equation is the key parameter that is temperature dependent in this equation. Its temperature dependency follows the classic Arrhenius relationship, which characterizes the temperature dependency of most chemical, physical, biochemical, and microbiological reactions. The Arrhenius equation for temperature dependency can be derived from a semi-log plot, in which the logarithm of the temperature dependent parameter is plotted against the reciprocal of absolute temperature, as indicated by equations (1.17)–(1.21): hC

C = C  · e Rs ·T

(1.17)

or ln C = ln C  +

hC Rs · T

(1.18)

By drawing an Arrhenius diagram with log C plotted against 1/T a straight line will result. From the slope of this straight line we can obtain the excess enthalpy hC term. The slope is: m=

hC 2.3 · Rs

where hC hvap hs,mono T Rs C, C 

(1.19)

specific bonding enthalpy of monolayer in J · kg−1 (desorption) specific vaporization enthalpy of adsorbate in J · kg−1 (desorption) specific sorption enthalpy of monolayers in J · kg−1 (desorption) absolute temperature in K specific gas constant of adsorbate in J · K−1 · kg−1 constants in BET equation

Therefore, the specific sorption enthalpy of the monolayer can be determined from the temperature dependency of the constants (C, C  ) in the BET equation. The “excess” monolayer bonding enthalpy hC disappears and plays no further role in the case of multilayer adsorption. In this case, other values of hC apply depending on the adhesive bonding of the second, third and additional multiple layers. With each additional layer, less and less bonding energy (enthalpy) is needed until the adsorbed water becomes freely available to behave as normal water (vaporize, condense, freeze, sublime and thaw), and only the enthalpy of vaporization (or condensation) applies.

1.2 Adsorption Equilibrium

15

1.2.6 Sorption of Water Vapor in Foods Much of the previous section was aimed at introducing many of the basic scientific principles of sorption phenomena in general. Although frequent reference was made to the processes of drying and rehydration (implying sorption of water) to help explain the distinction between desorption and adsorption, references to any specific substances were rarely made. Sorption processes can occur at the interface between solids and fluids (gases or liquids), or between fluids (gas–liquid, liquid–liquid). However, in the fields of food technology and engineering, an understanding of the sorption of water vapor in foods is of paramount importance in the design and specification of many food processing, packaging, storage and handling systems. In the case of water vapor adsorption (rehydration) in foods, we have sorption at a solid–fluid interface with water molecules being adsorbed onto the boundary surface of a dry solid material. Once monolayer saturation is achieved at the boundary surface and multilayers begin to form, the initially dry solid material will begin to absorb the water molecules through the porous structure into the interior of the material, just as a dry towel will absorb water from a wet surface. Thus, the initially dry solid material is called the “adsorbent.” When the process comes to an end at equilibrium, the resulting adsorbed water is called the “adsorbate” (see Figure 1.5). Just as in the previous section, the sorption of water vapor in foods is characterized by the sorption isotherms for water vapor in food materials, called vapor sorption isotherms. These vapor sorption isotherms are graphical plots showing the relationship between moisture content and water activity over a range of water activities at constant temperature for a given substance.For most food and biological materials the data points on these plots normally fall on a smooth curve of sigmoid shape, which becomes the sorption isotherm at that temperature for that substance. The precise shape and position of the sorption isotherm on the graphical plot is specific to each type of food material or substance, and can be considered a signature physical property of that substance. Figure 1.6 shows some examples of different sorption isotherms. Curve 1 shows a strong sorbent, which takes up large amounts of water although in a region of very low water activity or equilibrium relative humidity. After the rapid water uptake we see a plateau, which is the result of the monolayer that has formed becoming stable. Silica for example would have a curve of this

Figure 1.5. Sorption of water vapor to a solid food surface. 1: gaseous water molecules, 2: adsorbed water (adsorbate), 3: food (adsorbent)

16

1 Water Activity Figure 1.6. Examples of water vapor sorption isotherms. 1: silica (strong sorbent), 2: milk powder (multilayer forming material), 3: black tea (monolayer forming material), 4: fructose (nearly impervious to water, then rapid uptake of water until a syrup is formed)

shape. Curve 3 shows a similar behavior, but not with such extreme water uptake. When a material does not stop to adsorb water when a monolayer is formed and continues on to form second, third and additional multilayers without pausing,we see a shape like that shown in curve 2.Some materials such as PVC powder do not adsorb very large amounts of water, and they behave like a substance impervious to water. Curve 4 shows crystalline fructose which, in regions of lower water activity shows nearly impervious behavior, but on exceeding a certain level of water activity starts to dissolute and form a syrup. The isotherm in this region is a dashed line because the original sample (in the form of dry crystals) does not exist any longer. Examples of sorption isotherms for specific food and agricultural materials are given in Figure 1.7 (from [133]).

Figure 1.7. Examples of sorption isotherms: 1: dried peaches, 2: cotton, 3: soybeans, 4: wheat [133]

Note the sigmoid shape is similar to other isotherms described by the BET model. However, in the case of the classic isotherms discussed in the previous section, the plotted parameters were adsorbent mass on the vertical axis, and partial pressure on the horizontal axis. In the case of water vapor sorption isotherms in food system applications, these parameters are replaced by moisture content and water activity, respectively. Therefore, a full understanding of these two properties (water activity and moisture content) is needed before proceeding further.

1.2 Adsorption Equilibrium

17

1.2.7 Water Activity Recall that water activity was first mentioned in the introduction to this chapter as a property parameter that could be used to reflect the way in which water is bound to the internal structure of the food. The degree to which water is freely available to act as a solvent, to vaporize or freeze, or the degree to which it is chemically bound and unavailable can all be reflected by an ability to specify the water activity of a food material. Recall that a simple working definition for water activity was given at the beginning of this chapter as the equilibrium relative humidity of the food material when it has come into equilibrium with the relative humidity of the atmosphere surrounding it. Once this equilibrium is reached, the food sample neither gains nor loses moisture over time [133]. Recall further that the definition of relative humidity in a mixture of air and water vapor is the ratio of amount of water vapor in the mixture divided by the maximum amount of water vapor that could be held by the air at that condition (saturation). Mathematically, this ratio is the same as the ratio of water vapor partial pressure divided by saturation pressure, as expressed in equation (1.20): p = ' = aW (1.20) pS By substituting this definition in the BET equation,expression (1.21) is derived, and transformation of variables plotted on the graph in Figure 1.4, allows for the determination of the constants, as shown in the previous section. Thus, we can see how water activity (or equilibrium relative humidity) can take the place of partial pressure as the parameter plotted on the x-axis for obtaining water vapor sorption isotherms. 1 1 aW C−1 = · + · aW m 1 − aW ma · C ma · C

(1.21)

aW . Another possibility to Recall that the BET plot is a graph of versus m1 · 1−a W further simplify the use of water sorption isotherms, is to use the water content instead of m (mass of the water absorbed). The water content is what we know as moisture content of the food.

1.2.8 Moisture Content For general purposes, the moisture content of a food is normally expressed simply as the percent moisture in the food substance. Mathematically, this is the ratio of the mass of water contained in the food sample (adsorbent) over the total mass of food sample containing the moisture (adsorbate), expressed as a percent. However, moisture content used as the variable plotted on the vertical axis of vapor sorption isotherms often is expressed as the ratio of mass of

18

1 Water Activity Figure 1.8. Food material with 20% moisture (wb) and 25% moisture (db)

water mW (adsorbate) divided by mass of dry matter mdm , only (absorbent), as expressed in equation (1.22): mW (1.22) xW = mdm These two different methods for expressing moisture content in a food sample are known as“wet basis”(wb) and “dry basis”(db),respectively.The distinction between both methods of expression, as well as the ability to quickly calculate one from the other must be well understood. The illustration in Figure 1.8, shows a vertical bar representing a 100 g food sample composed of 20 g water and 80 g dry matter, and shows how these quantities are used to correctly express moisture content on either basis. For example, in this case the moisture content on a wet basis is 20/100 = 20%. But, on a dry basis it is 20/80 = 25%. Refer also to Table 1.6 for further elaboration on the difference between the two expressions and how to convert the quantities from one to another. Example 1.2. Moisture content of corn flakes (dry basis) A sample of corn flakes has a moisture content of 7.5% (wet basis). Express this moisture content on a dry basis. xW,db =

mW 7.5 kg 7.5 kg = = = 0.081 mtotal − mW 100 kg − 7.5 kg 92.5 kg

so xW,db = 8.1 g H2 O/100 g dm = 8.1% (db) or more simply xW,db

1 1 −1 0.075

= 0.081

1.2 Adsorption Equilibrium

19

Table 1.6. Conversion of moisture content on dry basis and wet basis dry basis (db) mW xW,db = mdm xW,db =

xW,db =

wet basis (wb) mW xW,wb = mtotal

mW mtotal − mW

xW,wb

mW mdm + mW

division by mW yields

1 1

xW,wb =

xW,wb = −1

1 1 xW,db

+1

possibilities to present the result xW (wb) xW (db) xW on dry basis xW on wet basis xW in g H2 O/g dry matter xW in g H2 O/g

Example 1.3. Moisture of corn flakes (wet basis): A sample of corn flakes has a moisture content of 21.2% (dry basis). Express the moisture content on a wet basis. mW 21.2 kg xW,wb = = = 0.175 mdm + mW 100 kg + 21.2 kg xW,wb = 17.5 g H2 O/100 g = 17.5% (wb) When we refer to moisture content in food sorption phenomena in this book, we always mean moisture content on a dry basis. 1.2.9 Hygroscopicity Hygroscopicity is a term used to describe how readily a material will take up moisture when subjected to a given shift (change) in relative humidity. The term is often used in industrial practice to indicate materials that quickly become problematic in the presence of the least increase in surrounding relative humidity. For example, certain powdered ingredients are said to be very hygroscopic if they become sticky and cause caking problems that prevent free flowing from the storage hopper in a food process. In this regard the term is used in a relative qualitative sense, and no quantitative scale has ever been assigned to hygroscopicity. We have no means to quantify what is the difference between strong and weak hygroscopic behavior. It may be possible to consider hygroscopicity as a material property that could be quantified if we understand that this type of moisture uptake behavior in response to change in relative humidity can be observed from the shape of the sorption isotherm for this material. For example, a highly hygroscopic powder will show a much higher uptake of water when shifted from 45% to 75% relative humidity than a powder would with lower hygroscopicity.

20

1 Water Activity Figure 1.9. Exposing four materials exhibiting different degrees of hygroscopicity to a shift in water activity from aW (A) to aW (B). The same shift causes different amounts of water uptake x

A

B

To help better understand this, consider the four different sorption isotherms shown in Figure 1.9. Each isotherm belongs to a different material, and all four materials exhibit different degrees of hygroscopicity. Let us assume that we shift the equilibrium relative humidity (water activity) from water activity aW (A) to water activity aW (B) on all four materials. Then, we can observe the different quantities of water uptake experienced by each material as an indicator of different degrees of hygroscopicity. It clearly can be seen that material 2 will have greater water uptake than material 1. The slope of the secant in the segment of the curve between the starting point and final point of the shift in water activity can be taken as a water uptake potential (wP -value) of the given material. The wP -value serves only as an indicator of the moisture difference x, which occurs on a given change aW . It does not take into account the level of moisture already absorbed into the material earlier. So, when we examine Figure 1.9 once again, we would have to note that although material 1 is a very good sorbent, when we make our shift in water activity aW (A) → aW (B) experiment, material 2 will show the higher water uptake. Therefore, we can say that material 2 has a greater wP than material 1 in the range of water activities of our experiment. When we use wP as an indicator of hygroscopicity, the points A and B have to be clearly specified. Also, a constant temperature was assumed throughout this example. 1.2.10 BET Equation for Foods In the case of water vapor sorption in foods, when the isotherms are obtained as plots of moisture content (db) versus water activity, the BET equation gets the form 1 aW 1 C−1 · = (1.23) + · aW xW 1 − aW xW,a · C xW,a · C

1.2 Adsorption Equilibrium

21 Figure 1.10. BET plot of adsorbed moisture content versus water activity

where aW xW C xW,a

water activity moisture content (db) in kg/kg dry matter BET constant monolayer moisture content (db) in kg/kg dry matter

aW If the quantity x1W · 1−a were plotted against water activity aW , a straight line W is obtained (see Figure 1.10). The monolayer moisture content xW,a as well as the BET constant, can then be obtained from the slope and intercept of the line. The value of xW,a is usually the moisture content at which the water is tightly bound with water molecules in a single monolayer, and it cannot participate as a solvent. Thus, it is the moisture content that should be reached for maximum stability of dehydrated foods (see also Section 1.3). The evaluation of the plot is performed again by obtaining the intercept and slope of the straight line curve in the diagram.

intercept: a =

1 xW,a · C

slope:

b=

C−1 xW,a · C

Because of ab = C − 1 or C = ba + 1 the moisture content of the food at the point where a complete monolayer exists is 

xW,a = a·

1 1 = b a+b +1 a

The monolayer moisture content xW,a and BET constant C are the only two parameters needed to describe the sorption isotherm for a given food product. Published tables or on-line data bases listing values of these parameters for various food stuffs can be found in references [7–9]. Also, sorption enthalpies needed for estimating energy requirements in the engineering design of drying or dehydration processes can be obtained from the Arrhenius temperature dependency of these parameters [10].

22

1 Water Activity

Example 1.4. BET – isotherm for a food product: The moisture content (dry basis) of the product in equilibrium with different relative humidity is listed below: Construct the BET plot from these data, and calculate the values for the monolayer moisture content xW,a and BET constant C. ' /% r.h. 0 10 20 30 50 75

xW /% (db) 0.0 2.5 4.3 5.2 8.3 18.8

First we prepare a table of values; then we draw the BET diagram: ' /% r.h.

xW /% (db)

xW (db)

aW

0 10 20 30 50 75

0.0 2.5 4.3 5.2 8.3 18.8

0 0.025 0.043 0.052 0.083 0.188

0 0.1 0.2 0.3 0.5 0.75

aW 1 − aW 0 0.11 0.25 0.43 1.0 3.0

1 aW xW 1 − aW – 4.4 5.8 8.3 12.0 16.0

For the slope, we get b = 19.0 and for the intercept a = 2.4. So the monolayer moisture content is 1 1 xW,a = = = 0.0467 a + b 2.4 + 19.0 and the BET constant C is 19.0 b + 1 = 8.92 C = +1 = a 2.4

Figure 1.11. Evaluation of Example 1.4

1.2 Adsorption Equilibrium

23

BET { One-Point Method For the purpose of rough estimation, the straight line BET plot can be approximated by constructing a straight line from the origin of the coordinate axes through a single data point. The single data point should be taken at value of water activity at which the monolayer is fully developed (saturated) but no multiple layer formation exists. As a rule of thumb for most foods, this value is normally chosen in the range of water activity between 0.3–0.4. The BET parameters for the monolayer moisture content xW,a and BET constant C can be estimated from the slope of this straight line, and will normally serve sufficiently well for most purposes of approximating the profile of the sorption isotherm for the given food substance, as shown by equation (1.24). Because of a = 0 we can say: xW,a =

1 b

(1.24)

and C=

b +1=∞ a

(1.25)

Example 1.5. Sorption isotherm of a food using the BET – one-point method: A food product has a moisture content of 5.2% (db) in equilibrium with air at a relative humidity of 30% r.h. Construct the one-point BET straight line, and estimate the monolayer moisture content xW,a for this food. With that single point we get the following: ' /% r.h.

xW /% (db)

xW (db)

aW

30

5.2

0.052

0.3

aW 1 − aW 0.43

1 aW xW 1 − aW 8.2

Therefore, without diagram we see the slope will be b = 8.2−0 0.3−0 = 27. The intercept was set to zero: a = 0. So the monolayer moisture content can be 1 estimated to be about xW,a = 1b = 27 = 0.037 Thermodynamics of Water Vapor Sorption in Foods Similar to the discussion on thermodynamics of sorption isotherms presented earlier in this chapter, the temperature dependency of the BET constant C can be obtained from an Arrhenius plot of log C over reciprocal absolute temperature T. The excess enthalpy of adsorption, or monolayer-bonding enthalpy hC , can be obtained from the slope m of the straight-line Arrhenius semi-log plot, as given by equation (12.6): m=− with

hC 2.3 · Rs

(1.26)

24

1 Water Activity

hC = hs,mono − hvap

(1.27)

respectively hs,mono = hvap + hC

(1.28)

where specific bonding enthalpy of H2 O monolayer in J · kg−1 (desorption) hC hvap specific vaporization enthalpy H2 O in J · kg−1(desorption) hs,mono specific sorption enthalpy H2 O monolayer in J · kg−1(desorption) T absolute temperature in K Rs specific gas constant of H2 O (461 J · kg−1 · K−1 ) The specific sorption enthalpy hs,mono will be found at the drying condition which leaves only the monolayer, but fully intact.At this point the monolayer is a complete layer of water molecules tightly bound to the boundary surface of the food material, and the enthalpy at this point marks the distinction between “free” and “bound” water in the food. As explained earlier, the BET theory covers a monolayer only and there is no regard for multilayers (like in the GAB model, see below). That means water in the second layer is treated like “free” water. So in the BET theory the “excess” monolayer bonding enthalpy hC disappears and plays no further role. In this case, water is freely available to behave as normal water (vaporize, condense, freeze, sublime and thaw), and only the enthalpy of vaporization (in the case of desorption), or condensation (in the case of adsorption) applies. In the opposite direction when the “excess” monolayer bonding enthalpy hC takes on high values, the bonding strength at the monolayer surface becomes very strong. Further water removal beyond this point through normal drying processes becomes very costly and time consuming. Therefore, it is of critical importance to identify this point of distinction with respect to the “excess” monolayer bonding enthalpy hC in order to design and specify optimum drying,storage and packaging conditions to assure long-term stability of dehydrated foods. A fast method for quickly estimating this “excess” monolayer bonding enthalpy hC is possible by comparing water activity of a food at different temperatures, and determining the temperature dependency of water activity for the given food. This can be done with the Clausius–Clapeyron equation [11]: ⎛ ⎞ hC ⎜ d ln aW ⎟ =− ⎝ 1 ⎠ Rs d T xW

(1.29)

or in approximation  log aW hC =− 1 2.3 · Rs  T

(1.30)

1.2 Adsorption Equilibrium

25

where specific bonding enthalpy of H2 O monolayer in J · kg−1 (desorption) hC hvap specific vaporization enthalpy H2 O in J · kg−1 (desorption) hs,mono specific sorption enthalpy H2 O monolayer in J · kg−1 (desorption) aW water activity T absolute temperature in K Rs specific gas constant of H2 O Example 1.6. Starch powder monolayer sorption enthalpy From two sorption isotherms taken at different temperatures we read the water activities at moisture content xW = 10% (db): # /◦ C 30 80

aW 0.25 0.45

We transform these data to #1 /◦ C

T1/K

30

303.15

#2 /◦ C

T2/K

80

353.15

1 −1 /K T1 3.299 · 10−3 1 −1 /K T2 2.832 · 10−3

aW1

log aW1

0.25

–0.602

aW2

log aW2

0.45

–0.347

From equation (1.29) we get hC = −2.3 · Rs ·

 log aW 1  T

or hC = −2.3 ·

log aW2 − log aW1 1 1 − T2 T1

so hC = −2.3 · 0.461 kJ · kg−1 ·

−0.34678 − (−0.6020) 2.832 · 10−3 K−1 − 3.299 · 10−3 K−1

that is hC = 579 kJ · kg−1 The result is a positive enthalpy. It means that have 579 kJ · kg−1 to be introduced into the material to overcome the sorption of the monolayer molecules. To bring the water molecules from the monolayer phase into the gaseous phase according to equation (1.28) we need the total enthalpy of hs,mono = 2200 kJ · kg−1 + 579 kJ · kg−1 = 2779 kJ · kg−1.

26

1 Water Activity

When we calculate a monolayer sorption enthalpy based on two points only (here 30 ◦ C and 80 ◦ C) we should not overestimate the precision of the result. Although in this example we got the information that the heat needed for desorption of the monolayer molecules is about 20 to 30% higher than for free water: hC 579 = = 0.26 hvap 2200 Recall that vapor sorption isotherms, as well as the BET model, provide information on the relationship between moisture content and water activity only in the region of relatively low moisture content. Food with intermediate moisture content can contain water adsorbed in multilayers. Because the BET model does not address multilayers, it is not valid in these regions of higher water activity, and the following GAB model is better suited for this purpose. 1.2.11 GAB Model The GAB model (after Guggenheim, Anderson, and deBoer [12]) is a semitheoretical multimolecular adsorption model intended for use over a wide range of water activity, and can be written as a three-parameter model: aW 1 C−1 (1.31) = + · aW m · (1 − k · aW ) ma · C · k ma · C where water activity aW m mass of adsorbate in kg k GAB correction factor (0.7–1.0) C Guggenheim constant ma mass of monolayer in kg The GAB isotherm equation is an extension of the two-parameter BET model which takes into account the modified properties of the adsorbate in the multilayer region and bulk liquid (“free” water) properties through the introduction of a third parameter k. If k is less than 1, a lower sorption is predicted than that by the BET model, and allows the GAB isotherm to be successful up to water activity of 0.9. The GAB equation reduces to the BET equation when k = 1. The constants in the GAB equation (k and C) are temperature dependent, and are the means by which we can extract information when we construct and refer to sorption isotherms at different temperatures. Thermodynamics of Sorption Phenomena Temperature dependency of the GAB model is approached in the same way as with the BET model, in that the Guggenheim constant is analogous to the BET constant with respect to being temperature dependent according to the Arrhenius relationship, as shown in equations (1.32), (1.33) and (1.34):

1.2 Adsorption Equilibrium

C = C · e

hC Rs ·T

27

(1.32)

respectively hC (1.33) Rs · T So, by drawing an Arrhenius plot with log C versus 1/T we can get the monolayer bonding enthalpy hC from the slope (same way as done in the BET model). hC (1.34) m= 2.3 · Rs Unlike the BET model, the GAB model takes into account the fact that when multimolecular layers begin to develop, adsorbed water molecules does not instantly behave as free water. But instead, bonding forces gradually weaken with increasing multilayers, such that vapor pressure is slightly reduced from that of pure water. This is taken into account by the parameter k, which is used as a multiplier coefficient that serves as a correction factor (with values between 0.7 and 1.0) in order to take this behavior into account.The BET model neglects this effect. From the temperature dependency of the GAB parameter k, we can determine the excess enthalpy from bonding of the multilayers: ln C = ln C  +

hk

k = k · e Rs ·T

(1.35)

So by making the Arrhenius plot log k versus 1/T from the slope m we can get this excess enthalpy hk . hk (1.36) 2.3 · Rs where hk is the mean bonding enthalpy of layers above the monolayer. When such a n-layer shall be removed from a surface the enthalpy hS,multi is necessary, which is the sum of enthalpy of vaporization and that bonding enthalpy hk, in other words, m=−

hk = hS,multi − hvap

(1.37)

where hS,multi mean specific sorption enthalpy of the water multilayer in J · kg−1 (desorption) hvap specific enthalpy of vaporization for water in J · kg−1 hk mean specific bonding enthalpy of the water multilayer in J · kg−1 (desorption) Rs specific gas constant for water in J · kg−1 K−1 T absolute temperature in K The GAB model is most appropriate when we deal with water activities above 0.4–0.5. A little disadvantage of this model is the increased mathematical complexity in dealing with a three-parameter equation compared to the BET model [14].

28

1 Water Activity

One can also use the expression for moisture content on a dry basis from equation (1.31) to further simplify the GAB equation into the form of equation (1.38), respectively (1.39): xW =

xW,a · C · k · aW (1 − k · aW ) (1 + (C − 1) · k · aW )

(1.38)

xW =

xW,a · C · k · aW (1 − k · aW )(1 − k · aW + C · k · aW )

(1.39)

so

where aW xW C k xW,a

water activity moisture content (db) in kg · kg−1 dry matter Guggenheim constant GAB correction factor (0.7–1.0) monolayer moisture content (db) in kg · kg−1 dry matter

Other workers [13] have transformed the GAB equation into a quadratic form, as shown by equations (1.40)–(1.43). aW = ˛ · aW 2 + ˇ · aW +  (1.40) xW By taking experimental data of xW versus aW and bringing them into a function like equation (1.40) we can determine the coefficients ˛, ˇ,  of the polynomial, and from it we can calculate the GAB parameters [15]:   k 1 (1.41) ˛= −1 xW,a C   1 2 ˇ= (1.42) 1− xW,a C 1 xW,a · C · k 1  2 1 xW,a = ˇ 2 − 4˛ ·   =

k=− C=

2 · ˛ · xW,a ˇ · xW,a + 1

1 xW,a ·  · k

(1.43)

(1.44) (1.45) (1.46)

With the aid of appropriate computer software, these parameters can be calculated within a few seconds for daily routine use of the BET or GAB models. With the use of such software and appropriate data input, food engineers and scientists can quickly determine the information needed to design and specify optimum processing, packaging, storage and handling conditions with respect to water activity requirements for shelf life stability of foods [16].

1.2 Adsorption Equilibrium

29

1.2.12 Other Models Several other models to mathematically describe the profile of sorption isotherms in foods have been proposed and reported in the literature [15,17]. They differ in regard to their physical approach, method for mathematical derivation,and number of parameters.Tables 1.7 to 1.9 contain a listing of these other models organized by the number of parameters required of each model. Models with only two parameters are listed in Table 1.7, three-parameter models are listed in Table 1.8, and four-parameter models in Table 1.9. Sometimes for special recipes other models can be applied, for example for sugar syrups or confectionery products there are special models in use. For more information and examples refer, e.g. to [134,135]. Table 1.7. Two-parameter models for sorption isotherms Freundlich (1906) Langmuir (1916) Smith (1947) Oswin (1946)

xW = A · aBW   aW xW = xmax aW + B xW = A + B · ln(1 − aW )  B aW xW = A 1 − aW

ln(1 − aW ) = −A

Henderson (1952)

xW

Brunauer, Emmet, Teller (1938) (BET)

xW =

 B1

xW,a · C · aW (1 − aW )(1 + (C − 1)aW )

1 C−1 aW 1 + · aW · = xW 1 − aW xW,a · C xW,a · C Halsey (1948)

 B1 A xW = − ln aW

Iglesias, Chirife (1978)

 ln aW 1 · ln − B A

1 2 ln xW + (xW + x0.5W ) 2 = A · aW + B

Lewicki (2000)

xW = A ·

Chung, Pfost (1967)

xW = −



1 −1 aW

B−1 −1

30

1 Water Activity

Table 1.8. Three-parameter models for sorption isotherms Cubic model GAB: Guggenheim (1966), Andersen (1946), De Boer (1953) expressed by Bizot as:

xW = p1 + p2 · aW + p3 · a2W + p4 · a3W xW =

xW,a · C · k · aW (1 − k · aW )(1 + (C − 1) · k · aW )

aW = ˛ · a2W + ˇ · aW +  xW

Table 1.9. Four-parameter models for sorption isotherms Peleg (1993)

xW = A · aCW + B · aDW

Isse (1993)

ln xW = A · ln

where xW aW xm x0.5m A, B, C, D

aW +B 1 − aW

moisture content (db) in kg · kg−1 dry matter water activity monolayer moisture content (db) in kg · kg−1 dry matter moisture content in kg · kg−1 (db) at aW = 0.5 Constants

1.3 Shelf Life of Food Related to Water Activity The normally high moisture content in most fresh foods is largely the reason they are so perishable. If left unprotected without being processed or preserved in any way, they will deteriorate rapidly as a result of various microbial, chemical and biochemical reactions. In order for these reactions to proceed, most of them require abundant availability of free water to act as a solvent and for hydraulic transport of molecules across semi-permeable membranes, essential for microbial metabolism, within the food [1]. For this reason one of the most effective methods of food preservation is to reduce the moisture content until the water activity is low enough that the amount of free water needed to support any of the degradation reactions that might take place is not available. This is the principle behind the storage stability and long shelf life of dehydrated foods. As the water activity is brought below 1 and lowered further in a food, the rates of these reactions begin to slow down. They proceed more slowly with further lowering water activity, which translates into longer and longer shelf life. The actual shelf life of any specific food product at a given water activity may vary depending on structure and composition of the food material, and spoilage mechanism of concern. The shelf life of a food will be limited by any one of a series of spoilage mechanisms that can destroy the food. These include microbial activity (bacteria, yeasts and moulds), enzyme activity, browning reactions, and lipid oxidation (rancidity). Of all these reactions, microbial

1.3 Shelf Life of Food Related to Water Activity

31

Table 1.10. Minimum water activity for population growth [109] aW 0.91–0.95 0.88 0.80 0.75 0.70 0.65

lower limit for growth of most bacteria most yeasts most moulds halo-tolerant bacteria osmo-tolerant yeasts xero-tolerant moulds

Figure 1.12. Relative rate (Vrel ) of different spoilage reactions as a function of water activity aW in food. 1: lipid oxidation, 2: browning reactions, 3: enzymatic reactions, 4: moulds, 5: yeasts, 6: bacteria. The dashed line indicates the sorption isotherm of the sample material

activity is by far the most sensitive to water activity, having the greatest need for available free water. Table 1.10 lists the minimum values of water activity needed for the growth of different microorganisms. The relative rate of activity of these various spoilage reactions (microbial, enzymatic, browning, oxidation) as a function of water activity is portrayed in Figure 1.12 after Labuza [18]. Wide reference is made to this diagram by food engineers and scientists for the purpose of estimating the range of water activity that may be needed to prevent a given spoilage reaction [14–19]. Table 1.11 shows typical values of water activity for various food products: It can now be apparent how all the different relationships discussed in this chapter begin to fit together. Water activity dictates whether or not a reaction will take place in a food, and if so, at what rate. Therefore the objective of most food dehydration processes is to bring the food product to a specified water activity. However, the engineer responsible for the design and operation of the food dehydration process can only be concerned about moisture content. The food drying equipment is designed only to remove moisture, and only moisture content can be measured and controlled on the factory floor (not water activity). The sorption isotherm becomes the fundamental tool by which it is possible to specify the moisture content needed that will assure the required water activity.

32

1 Water Activity

Table 1.11. Typical water activities of food (examples, average data), from [109] and [131] aW ≈ 0.99 ≈ 0.95 ≈ 0.8–0.9 ≈ 0.7–0.8 ≈ 0.6–0.7 ≈ 0.5 ≈ 0.4 ≈ 0.3 ≈ 0.2

milk, meat, fish, juice, fruits, vegetables sausage, liverwurst, camembert ketchup, jam, bread, syrup, salami prunes, marzipan honey, raisins, nuts, dried fruits bakery products, spices egg powder cookies milk powder

Figure 1.13. Generic sorption isotherm showing main regions of water bonding

The shape of the sorption isotherm also helps us to know what to expect about the quality of physical characteristics of a food product as a function of water activity. For example, we often divide a sorption isotherm into the three regions of water bonding states, as shown in Figure 1.13. Region I is the region of very lowest water activity and moisture content. In this region, the very small amount of water in the material is completely bound within the monolayer, and unavailable to contribute any kind of moisture behavior to the material. Most food materials would be completely shrunken, hard and brittle in this state. It is also very difficult and costly to attempt to remove sufficient water by drying processes to reach such low levels of water activity. For these reasons, we often avoid to bring foods into Region I of their sorption isotherms. Region II is the region in which the water activity is sufficiently low that most food spoilage reactions cannot proceed. This is the region in which the monolayer is completely saturated, but no free moisture is available for supporting any chemical, biochemical or biological activity. Therefore, this is the region into which we bring dehydrated foods for long term storage stability, such as food powders and dry snacks. These dry foods normally have a crispy or crunchy texture.

1.4 Laboratory Determination of Sorption Isotherms

33

Region III is the region in which moisture uptake occurs most rapidly as multilayers are formed and the moisture content reaches the levels in which the water can act freely once again to support chemical and biological reactions. This is also the region in which the food material will exhibit the textural properties of being “fresh” with soft to firm elastic properties while being moist and swollen (fresh fruit and vegetables). Some foods called “intermediate moisture” foods are dehydrated to a carefully controlled intermediate level of water activity in the region of transition between Regions II and III. The precise level of water activity is chosen to be sufficiently low so that no free moisture is available for supporting any spoilage reactions, but sufficiently high to still give a soft flexible texture. These foods are ready to eat and require no rehydration (beef jerky,salami,etc.).At the same time they are shelf stable for long periods of storage just as any dehydrated foods.

1.4 Laboratory Determination of Sorption Isotherms In order to obtain a sorption isotherm for any material substance,it is necessary to be able to accurately measure both the moisture content and the water activity of a sample after it has come into equilibrium relative humidity at different known levels of relative humidity. The moisture content must be on a dry basis (db), as described in Table 1.6. Measurement of Water Activity Different procedures are available for measuring water activity [20], but they usually fall into one of three general methods of approach. Water activity can be measured through:

• Sample being brought into equilibrium with a closed atmosphere of known constant relative humidity (isopiestic technique or desiccators method) • Equilibrium head space atmosphere surrounding sample is measured for its relative humidity with the sample (water activity meter) • Dynamic method in which samples are exposed to atmospheres of various relative humidity and weighed simultaneously. The first method of approach involves samples of known initial moisture content which go through the process of moisture gain (adsorption) or loss (desorption) in a constant controlled relative humidity atmosphere. The sample weight will change over time as it gains or loses moisture, and is measured and recorded periodically until the sample has come into equilibrium with the relative humidity of the surrounding atmosphere. At this point, when the sample will neither gain nor lose moisture with further time, the sample will have a water activity equal to the controlled constant relative humidity used

34

1 Water Activity

throughout the experiment therefore this approach sometimes is called isopiestic technique. The second approach is to place the food sample of known constant moisture content into a sealed enclosure with small head space. The atmosphere in the head space is then probed for measurement and monitoring of changes in relative humidity or partial pressure until equilibrium is reached. The relative humidity measured at equilibrium is taken as the water activity of the sample. Because time to reach equilibrium can be long e.g. several hours, some instruments are equipped with software for extrapolation to the water activity at which the sample will neither gain nor lose moisture. In the third approach (dynamic method), samples of food can be exposed for a short period of time to flowing air at various relative humidity and weighed automatically. For this purpose, isothermal thermogravimetric methods are useful techniques (see Section 7.9.1). Regardless of the different approaches, it is important in all three methods to remember that sorption is strongly temperature dependent, and the temperature must be kept constant and controlled while all the experiments are being carried out. It is also important to determine and note the distinction if the experiment is for the purpose of constructing a desorption (drying) or adsorption (wetting) isotherm, because of hysteresis effects explained earlier. In the case of a desorption isotherm, the food sample will initially be very fresh with relatively high moisture content (fully hydrated), and will lose moisture and weight during the experiment. In the case of an adsorption isotherm, the food sample will initially be very dry (dehydrated), and will gain moisture and weight during the experiment. Measurement of Moisture Content The standard classic procedure for accurate determination of the moisture content in a sample of food material is the gravimetric oven method [103]. In this method, a sample of known initial weight is placed into a drying oven at 103 ◦ C, and weighed periodically until no further weight loss is detected. There are various types of laboratory “moisture meters” available for more rapid determination of moisture content (within 10–30 minutes). These work on the principles of either measuring electrical conductivity (see, e.g. [110]), which depends on moisture content, or by driving the moisture out of the sample more rapidly than a drying oven using infrared radiation from an infrared lamp. These methods, however, can often give erroneous results because of various limitations, and must be used with great caution. Laboratory Apparatus for Determination of Sorption Isotherms Of all the methods described earlier for measuring water activity, the “desiccator jar” method in which the sample is brought into equilibrium in a closed system with a known constant relative humidity, is considered to be the classic

1.4 Laboratory Determination of Sorption Isotherms

35

Figure 1.14. Taking a point of the sorption isotherm. The sample is placed in a closed chamber with a humidity standard (at the bottom, schematic). After water activity equilibration of sample and atmosphere in the chamber the sample is weighed to get its moisture content

standard for the purpose of constructing sorption isotherms with the greatest accuracy. In order to construct a sorption isotherm several data points are needed, each point defined by the moisture content (db) measured at a specified water activity, over a range of specified water activities. Samples are brought to specified water activities by placing them in hermetically sealed desiccator jars, shown in Figure 1.14. Each jar maintains a sealed headspace atmosphere around the sample at known constant relative humidity. Samples are then periodically weighed to monitor weight gain or loss until equilibrium is reached when no further weight gain or loss occurs over time. The covers on these jars are designed in such a way that they can be easily opened and re-closed quickly for removal and replacement of the samples during the few minutes required for periodic weighings. More sophisticated desiccator jar systems allow for continuous monitoring of weight change by having the samples resting upon electronic load cells within the jar, so there is no need to open the jar during the experiment.The relative humidity in jar head space can be measured accurately by means of a hygrometer. The time period between sample weighings will depend upon how rapidly equilibrium will be reached. The time required to reach equilibrium at any given relative humidity can vary widely depending upon the size and shape as well as physical and chemical make-up of the food sample. Sometimes this can be a problem with many fresh food samples that are highly perishable [21, 35]. They may begin to deteriorate as a result of microbial spoilage activity before equilibrium is reached. This would compromise the validity of the data. In a laboratory experiment for determining a sorption isotherm on a food sample, several desiccator jars are needed, one for each data point that will be used to construct the isotherm with each jar maintaining a different controlled relative humidity. The method most commonly chosen for establishing a different relative humidity in each jar is to place within each jar a saturated solution of a given salt. Saturated salt solutions are frequently used as relative humidity standards. Table 1.12 shows some salts frequently used for this purpose, and their water activities (equilibrium relative humidities) at different temperatures. Another isopiestic technique is to bring a droplet of an indicator liquid into water activity equilibrium with the sample. The sample is placed in a small jar and a small amount of the indicator liquid is placed in glassware in the same jar. Now the indicator fluid will take up water from the sample head space

36

1 Water Activity

Table 1.12. Water activity standards saturated aqueous solution of:

water activity aW at

from 10 ◦ C –

15 ◦ C 20 ◦ C 25 ◦ C 30 ◦ C 35 ◦ C [37] 0.0957 0.0891 0.0824 0.0758 0.0692 ±0.028 ±0.024 ±0.021 ±0.017 ±0.015 [37] 0.1129 0.1130 0.1131 0.1130 0.1128 0.1125 ±0.0041 ±0.0035 ±0.0031 ±0.0027 ±0.0024 ±0.0022 [134] – – 0.119 – – –

NaOH Sodium hydroxide LiCl Lithium chloride ZnCl2 Zinc chloride K-CH3 CO2 [37] 0.2338 0.2340 0.2311 Potassium acetate ±0.0053 ±0.0032 ±0.0025 MgCl2 [37] 0.3347 0.3330 0.3307 Magnesium chloride ±0.0024 ±0.0021 ±0.0018 K2 CO3 [37] 0.4314 0.4315 0.4316 Potassium carbonate ±0.0039 ±0.0033 ±0.0033 MgNO3 [37] 0.5736 0.5587 0.5438 Magnesium nitrate ±0.0033 ±0.0027 ±0.0023 NH4 NO3 – – 0.66 Ammonium nitrate NaCl [134] 0.7567 0.7561 0.7547 Sodium chloride ±0.0022 ±0.0018 ±0.0014 KCl [37] 0.8677 0.8592 0.8511 Potassium chloride ±0.0039 ±0.0033 ±0.0029 KNO3 [37] 0.9596 0.9541 0.9462 Potassium nitrate ±0.014 ±0.0096 ±0.0066

0.2251 ±0.0032 0.3278 ±0.0016 0.4316 ±0.0039 0.5289 ±0.0022 0.66

0.2161 – ±0.0053 0.3244 0.3205 ±0.0014 ±0.0013 0.4317 – ±0.0050 0.5140 0.4991 ±0.0024 ±0.0029 – –

0.7529 0.7509 0.7487 ±0.0012 ±0.0011 ±0.0012 0.8434 0.8362 0.8295 ±0.0026 ±0.0025 ±0.0025 0.9358 0.9231 0.9079 ±0.0055 ±0.0060 ±0.0083

until equilibrium is reached. Then the indicator fluid is taken out and analyzed by a refractometer for its water content. Having the sorption isotherm of the indicator fluid (water content versus its water activity) the water activity of the sample can be determined very simply and quickly [22]. Note that a high quality constant-temperature refractometer is needed for this procedure. It is also important to note that sample size has a great effect on time required to reach equilibrium because of time needed for diffusion of molecules through the interior tissue structure of the sample. Laboratory methods and procedures that permit very small sample sizes, have the advantage that they are likely to produce valid results far more quickly [36]. Standards for Measuring Sorption Isotherms In order to validate the accuracy of any given laboratory procedure or analytical method for measuring a physical or chemical property, most laboratories

1.4 Laboratory Determination of Sorption Isotherms

37

rely upon testing their procedures with a “standard” sample to see if the result is in accordance with the expected known outcome for the standard. In the case of water vapor sorption isotherms in foods, the recommended standard is microcrystalline cellulose (MCC). This material is widely available to the scientific community under the trade name “Avicel,” and possesses a signature isotherm,that is published in books on physical properties of foods [23].Figure 1.15 shows the sorption isotherm for this standard material. Equation (1.47) gives the mathematical model for this isotherm when the numerical value of the constants A and b are taken as 0.0540 and –0.4343, respectively (Avicel PH1 at 25 ◦ C). b  1 (1.47) xW = A aW − 1 The following table compares model predicted with experimentally measured moisture content of a sample of Avicel. Table 1.13. Predicted and measured data of Avicel aW 0.1 0.2 0.4 0.6 0.8 0.9

xW in % (db) predicted 2.80 2.96 4.53 6.44 9.86 14.02

xW in % (db) measured (from [23]) 2.0 ± 0.5 3.0 ± 0.5 5.0 ± 0.5 7.0 ± 0.5 10.0 ± 0.5 –

Figure 1.15. Sorption isotherm of microcrystalline cellulose (25 ◦ C, adsorption) (from [23])

38

1 Water Activity

1.5 Applications sugar and sugar replacers: phase transitions by water uptake lupinus: sorption isotherms and monolayer bonding enthalpy potato: sorption isotherms and monolayer bonding enthalpy starch powders: sorption isotherms agar gels: sorption isotherms, monolayer bonding enthalpy, Henderson model strawberries: GAB sorption isotherm and glass transition poultry meat: sorption isotherm and glass transition pineapple: sorption isotherm, sorption enthalpy pepper: sorption enthalpy starch and wheat gluten: GAB sorption isotherms pineapple: water desorption thermodynamic properties corn starch: modeling of water sorption isotherm

[24] [25] [26,27] [28] [29] [30] [31] [32] [33] [34] [39] [40]

Literature 1. Lewicki PP (2004) Water as the determinant of food engineering properties. A review. J Food Engineering, 61:483–495 2. Gregg SJ, Sing KSW (1982) Adsorption, Surface Area and Porosity. Academic Press, London 3. Luy B (1991) Vakuum Wirbelschicht. Dissertation, Universit¨at Basel 4. Korte F (ed) (1973) Methodicum chimicum Bd 1 Teil 2. Thieme, Stuttgart 5. Brunauer S, Emmet PH, Teller E (1938) J Am Chem Soc 60:309 6. Bauer KH, Fr¨omming KH, F¨uhrer C (1999) Pharmazeutische Technologie. Deutscher Apotheker Verlag, Stuttgart 7. Cazier JB, Gekas V (2001) Water activity and its prediction: a review. Intern J Food Properties 4:35 8. Iglesias HA, Chirife J (1982) Handbook of Food Isotherms: Water Sorption Parameters for Food and Food Components. Academic Press, New York 9. Nesvadba P, Houska M, Wolf W, Gekas V, Jarvis D, Sadd PA, Johns AI (2004) Database of physical properties of agro-food materials. J Food Engineering 61: 497–503 10. Taylor AJ (1998) Physical chemistry of flavour. Int J Food Sci Tech 33:53–62 11. Atkins PW (1990) Physikalische Chemie. VCH Weinheim 12. Van den Berg, Bruin S (1981) Water activity and its estimation in food systems. In: Rockland LB, Stewart GF (eds) Water Activity: Influence on Food Quality. Academic Press, New York 13. Bizot H (1983) in: Jowitt R, Escher F, Hallstr¨om B, Meffert HFTh, Spiess WEL, Vos G (1983) Physical Properties of Food. Applied Science Publishers, Barking 14. Timmermann EO, Chirife J, Iglesias HA (2001) Water sorption isotherms of foods and foodstuffs: BET or GAB parameters? J Food Engineering 48:19–31 15. Isengard HD, R¨uckold S, Grobecker KH (2001) Water as a source of errors in reference materials. Fresenius J Anal Chem 370:189–193 16. Labuza Th, University of Minnesota, available online from: http://www.fsci.umn.edu/Ted Labuza/CV/tplcv.htm [cited 2003–12–24]

Literature

39

17. Lewicki PP (2000) Raoult’s law based food water sorption isotherm. J Food Engineering 43:31–40 18. Labuza TP (1971) Kinetics of lipidoxidation in foods. Crit Rev Food Technol 2:355 19. Troller JA, Christian JHB (1978) in: Water Acivity and Food. Academic Press, New York 20. Hofer AA (1962) Zur Aufnahmetechnik von Sorptionsisothermen und ihre Anwendung in der Lebensmittelindustrie. Dissertation, Universit¨at Basel 21. Chirife J, Buera MP (1995) A critical review of some non-equilibrium situations and glass transitions on water activity values of foods in the microbiological growth range. J Food Engineering 25:531–552 22. Steele RJ (1987) Use of polyols to measure equilibrium relative humidity. Int J Food Sci Technol 22:377–384 23. Spiess (1983) in: Jowitt R, Escher F, Hallstr¨om B, Meffert HFTh, Spiess WEL,Vos G (1983) Physical Properties of Food. Applied Science Publishers, Barking 24. Cammenga HK, Gehrich K (2003) Glatt, rau oder klebrig? Zeitschrift f¨ur Lebensmittelund Verpackungstechnik (LVT) 48:28 25. V´azquez G, Chenlo F, Moreira R (2003) Sorption isotherms of lupine at different temperatures. J Food Engineering 60:449–452 26. McMinn WAM, Magee TRA (2003) Thermodynamic properties of moisture sorption of potato. J Food Engineering 60:157–165 27. McLaughlin CP, Magee TRA (1998) The determination of sorption isotherm and the isosteric heats of sorption for potatoes. Journal of Food Engineering 35:267–280 28. Al-Muhtaseb AH, McMinn WAM, Magee TRA (2004) Water sorption isotherms of starch powders: Part 1: mathematical description of experimental data. J Food Engineering 62:135–142 29. Iglesias O, Bueno JL (1999) Water agar–agar equilibrium: determination and correlation of sorption isotherms. Int J Food Sci Tech 34:209–216 30. Moraga G, Mart´ınez-Navarrete N, Chiralt A (2004) Water sorption isotherms and glass transition in strawberries: influence of pre-treatment. J Food Engineering 62:315–321 31. Delgado AE, Sun DW (2002) Desorption isotherms and glass transition temperature for chicken meat. J Food Engineering 5:1–8 32. Hossain MD, Bala BK, Hossain MA, Mondol MRA (2001) Sorption isotherms and heat of sorption of pineapple. J Food Engineering 48, 2:103–107 33. Kaymak-Ertekin F, Sultanoglu M (2001) Moisture sorption isotherm characteristics of peppers. J Food Engineering 47:225–231 34. Viollaz E, Rovedo CO (1999) Equilibrium sorption isotherms and thermodynamic properties of starch and gluten. J Food Engineering 40:287–292 35. R¨uckold S, Grobecker, KH, Isengard, HD (2001) Water as a source of errors in reference materials. Fresenius J Analytical Chemistry 370:189–193. 36. Czepirski L, Komorowska-Czepirska E, Szymonska J (2005) Adsorptive properties of biobased adsorbents. Adsorption 11:757–761 37. Greenspan L (1977) Humidity fixed points of binary saturated aqueous solutions. Journal of Research of the National bureau of Standards A: Physics and Chemistry, Vol 81A:89 38. DIN ISO 9277 (2003) Determination of the specific surface area of solids by gas adsorption using the BET method, in [101] 39. Simal S,Femenia A,Castell-PalouA,Rossell´o C (2007)Water desorption thermodynamic properties of pineapple. J Food Engineering 80:1293–1301 40. Guilan Peng, Xiaoguang Chen, Wenfu Wu, Xiujuan Jiang (2007) Modeling of water sorption isotherm for corn starch. J Food Engineering 80:562–567

2 Mass and Density

2.1 Mass Mass is a measure for inertia and heaviness of a body. Heaviness is caused by the Earth’s gravitational attraction for a body. The force between the body of interest and the planet Earth is called the weight force of the body. Mathematically, this force can be expressed as the product of the body’s mass and the Earth’s acceleration due to gravity, as shown by equation (2.1). G = m·g

(2.1)

where G weight force in N m mass in kg g gravitational acceleration in m · s−2 Because the density of planet Earth varies with location and the planet is slightly pear-shaped and not in the shape of a perfect sphere, the value of gravitational acceleration differs slightly with location on the Earth’s surface. Considering the rotation of the planet, a body resting at the equator will have a greater tangential speed and centrifugal force than in regions far north or south of the equator.The value of Earth’s gravitational acceleration in Zurich,Switzerland is used as a standard for calculations, and is called standard gravitational acceleration having the value g = 9.80665 m · s−2 [107]. When a balance which was adjusted in Zurich, is taken to another place on the Earth, but is not corrected for the local gravitational acceleration, the displayed weight may be in error.Table 2.1 illustrates this concept for a body having a mass of 1 kg.To avoid erroneous weight measurements of this type, a balance has to be recalibrated at the location in which it will be used. For this purpose, commercial mass standards are produced with the help of national standards organizations around Table 2.1. Weighing a 1 kg mass in different places City Zurich, Switzerland Bogota, Columbia Reykjavik, Iceland

g in m · s−2 9.80665 9.77390 9.82265

balance displays 1000.0 g 996.7 g 1001.6 g

42

2 Mass and Density

Table 2.2. Metrological institutes, examples ISO NIST

International Organization for Standardization National Institute of Standards and Technology (USA) PTB Physikalisch-Technische-Bundesanstalt (Germany) AIST National Institute of Advanced Industrial Science and Technology (Japan) NPL National Physical Laboratory (UK) IEN Istituto Elettrotecnico Nazionale Galileo Ferraris (Italy) NRC National Research Council (Canada) METAS Bundesamt f¨ur Metrologie und Akkreditierung (Switzerland) CSIRO Division of Telecommunication & Industrial Physics (Australia) NML National Metrology Laboratory (South Africa) NMi Nederlands Meetinstituut (Netherlands)

www.iso.org www.nist.gov www.ptb.de www.aist.go.jp www.npl.co.uk www.ien.it http://inms-ienm.nrc-cnrc.gc.ca www.metas.ch www.csiro.au www.csir.co.za www.nmi.nl

the world. These organizations assume responsibility for control and uniformity of standards from different countries. For example the national 1 kg mass standards that are used throughout Europe are copies of a platinum iridium prototype, which is kept at BIPM (Bureau International des Poids et Mesures) in Sevres near Paris, France. Similar organizations/institutes responsible for national standards in other parts of the world are identified in Table 2.2.

2.2 Weighing and Atmospheric Buoyancy A balance is an instrument measuring the weighing force of a body. However, it usually does not display a force signal (e.g. newtons), but a mass signal (e.g. kilograms). This is due to the principle of calibration used for balances: A mass standard is placed on the balance that causes a deformation, which can be read as an angle, a distance or an electric voltage, depending on the type of balance. With different mass standards a mass versus reading data set is obtained. This can be used to generate a scale to be mounted on the balance. With that scale in place, the balance operates as though it were an instrument for measuring mass. This procedure is called a calibration. A calibration has to be performed for every type of sensing/measuring instrument. For this purpose, appropriate respective standard materials and procedures are needed, that make it possible to perform calibration of an instrument in any laboratory. However, in many parts of the world, calibration of balances or check weighers, that are to be used in commerce (sale of products) often has to be done by official institutions with official standards depending on national laws.

2.2 Weighing and Atmospheric Buoyancy

43

From a scientific point of view the calibration procedure described for a balance is basically to use the instrument as a “force meter,” then divide the force G measured by the value of the local gravitational acceleration g, and display the result (equation volume of hydrometer in m3 ). G =m (2.2) g Since the middle ages the weight of a body has been a manifold of a reference weight.So weighing is simply a comparison to a given mass standard.From this point of view weighing is dividing the weight force of the given body and weight force of a mass standard and the result is a dimensionless number. That is the principle of all mechanical and electronic balances up to today, and a consequence of lacking an expression for mass with fundamental natural constants. GK mK · g = =n (2.3) GR mR · g where GK GR mK mR n g

weight force in N weight force of mass standard in N mass of body in kg mass of mass standard in kg ratio gravitational acceleration in m · s−2

Most weight measurements are carried out with body and balance surrounded by atmospheric air, which is a gaseous fluid possessing density. Only bodies of material with density greater than atmospheric air at the Earth’s surface can impart a force when placed upon a balance. For example a rubber balloon filled with helium gas (less dense than air) possesses mass, but it will not rest on a balance. It will rise upward into the atmosphere in search of an altitude at which the density of the atmosphere is in equilibrium with itself. His upward force caused the density of the Earth’s atmosphere is known as buoyancy. This atmospheric buoyancy causes a body resting on a balance when surrounded by atmospheric air to exhibit a slightly smaller weight measurement than if it were in a vacuum (Figure 2.1).

Figure 2.1. A balance is adjusted to zero weight (picture I). A mass of 1 kg is weight in atmosphere (picture II) and in vacuum (picture III)

44

2 Mass and Density

The calculation needed to correct for this buoyancy effect in order to yield the true mass of a body is called atmospheric buoyancy correction. The true mass of a body mK is the product of the displayed mass m∗K and a correctional factor K. The value of the correctional factor K depends on the density of the air surrounding the balance. Because weight measurement is a comparison of a body of interest and a mass standard, the densities of both materials also influence the correctional factor (equations (2.4) to (2.6): mK = m∗K · K

(2.4)

1 1 − K R K = 1 + L · L 1− K   1 1 K ≈ 1 + L · − K R where m∗K mK L K R K

(2.5)

(2.6)

displayed weight of the body in kg true mass of the body in kg density of air in kg · m−3 density of the body in kg · m−3 density of mass standard in kg · m−3 correctional factor for atmospheric buoyancy

The density of air depends on its pressure, temperature, humidity and its concentration of CO2. Values are listed (see, e.g. [105]) or can be approximated with empirical equations (see e.g. [106]). For practical purposes, the density of atmospheric air at normal room temperature and sea level (standard conditions) can be taken to be approximately 1.2 kg · m−3 . A simple approach to calculate the density of air more precisely (on the basis of [1]) is given by equation (2.7):   p/Pa p∗ (2.7) L /kg · m−3 = 3.4849 · 10−3 · 1 − 0.3780 T/K p where p air pressure in Pa p∗ water vapor partial pressure in Pa T temperature in K Example 2.1. Estimation of air density air pressure air temperature air humidity water vapor partial pressure

p = 105 Pa # = 20 ◦ C i.e. T = 293.15 K ' = 50% r.h. p∗ = ' · ps = 0.5 · 23.27 Pa = 11.7 Pa

2.3 Density

45

Table 2.3. Atmospheric buoyancy correctional factor, examples food material cocoa butter water sucrose

/kg · m−3 915 1000 1590

K 1.00116 1.00105 1.00091

m∗K /g 1000.00 1000.00 1000.00

mK 1001.16 1001.05 1000.91

difference 0.12% 0.11% 0.09%

Values of water vapor pressure ps are listed in water vapor steam tables, see, e.g. Appendix 15.8.   5 11.7 −3 −3 10 L /kg · m = 3.4849 · 10 · 1 − 0.3780 · 5 = 1.1887 293.15 10 so L = 1.189 kg · m−3 In contrast to air, the density of water is 1000 times greater (1000 kg · m−3 ). Therefore, the density range of most food, agricultural and biological materials is in the same order of magnitude as that of water, about 1200 ± 300 kg · m−3 . The densities of materials with high water content are in a range more closely to that of water (between 1000 and 1100 kg · m−3) while dry materials like agricultural grains, seeds and dry beans consisting of proteins, carbohydrates, starch or cellulose are often in the range 1400–1600 kg · m−3 . Since these density ranges are one thousand times greater than the density of air, the correctional factor for atmospheric buoyancy can be estimated to be approximately 0.1%. Table 2.3 shows some examples calculated with an air density of L = 1.2 kg · m−3 and density of a mass standard used for calibration of the balance of R = 8000 kg · m−3 . Note that the density of the mass standard used in calibration is far greater than the range of most food materials because these mass standards are fabricated from very dense metals (brass, stainless steel, chromium, etc.). Because of the small effect of atmospheric buoyancy in most food technology applications, the systematic error of atmospheric buoyancy is frequently neglected. Often this error is overshadowed by other factors, such as uncertainty of weighing accuracy and lack of precision, which may be larger. However, for the fabrication of mass standards to be used in calibration, and in commercial trade of highly expensive materials it must be taken into account.

2.3 Density Frequent reference to the physical property of density was made in the previous subsection with the expectation that the reader would have an intuitive “operational definition“ of density. In this subsection, a more scientific definition is presented. The density of a substance is the quotient of mass over volume. The standard international (SI) units for expressing density are kg · m−3 . The same

46

2 Mass and Density

definition is valid for solid, liquid, gaseous and disperse systems like foams, bulk goods or powders. The reciprocal of density is called specific volume and the units are m3 · kg−1 (equations (2.8) and (2.9). m = (2.8) V 1 V =v= (2.9) m  where m V  v

mass in kg volume in m3 density in kg · m−3 specific volume in m3 · kg−1

2.3.1 Temperature Dependency of Density Many materials undergo thermal expansion when heated, meaning they increase in volume without any change in mass. For this reason, the density of a given material often depends on temperature. Since the volume of a material normally increases with temperature, the density usually decreases with temperature. This effect is much larger in gaseous systems than in liquid or solid systems. Ideal Gases For many engineering applications air can be assumed to behave as an ideal gas, meaning that the ideal gas law can be used for calculating the density of air as a function of temperature and pressure (equations (2.10) and (2.11). p · V = m · RS · T = where p Rs  T m V

p RS T

(2.10) (2.11)

pressure in Pa specific gas constant in J · K−1 · kg−1 density kg · m−3 temperature in K mass in kg volume in m3

In case of low temperatures and humid air the ideal gas law loses accuracy, and will lead to error. To calculate the density of air more precisely as a function of water vapor partial pressure and atmospheric pressure, equation (2.7) can be used.

2.3 Density

47

Solids and Liquids The density of liquids and solids is a function of temperature. Small changes in volume caused by temperature change can be calculated with the aid of the thermal expansion coefficient:  =

1 dV · V dT

(2.12)

where  thermal expansion coefficient in K−1 T temperature in K V volume in m3 For numerous materials, values of density at different temperatures are published in tables (e.g. for air [105, 106], water [105, 108], milk [109]), or can be calculated with polynomial functions (see Section 15.9). Water shows abnormal behavior in a narrow range of temperature near its freezing point at atmospheric pressure. When lowering the temperature of water from 4 ◦ C and 0 ◦ C, the density of water actually decreases rather than increases. This abnormal behavior of water (see Figure 2.2 and Figure 2.3 is taken into account within the polynomial function of Bertsch (1983) for calculation of the density of liquid water [106] (equation (2.13): /kg · m−3 = 1000.22 + 1.0205 · 10−2 · (# /◦ C) − 5.8149 · 10−3 · (# /◦ C)2 + 1.496 · 10−5 · (# /◦ C)3

(2.13)

A further abnormality of water is that the solid phase (ice) has a lower density than the liquid phase at the same temperature. This behavior has important consequences for the biosphere. Because of this temperature dependency of density, control or measurement and recording of temperature is necessary when density is measured for

Figure 2.2. Normal (N) and abnormal (H2 0) thermal expansion (schematic)

Figure 2.3. Abnormality of water (H2 O): Temperature dependency of density compared to normal behavior (N)

48

2 Mass and Density

process control and quality control purposes (e.g. in-line sensors during food processing, see Chapter 14). 2.3.2 Pressure Dependency of Density Materials are compressible. On application of pressure their volume decreases, causing the density to be a function of pressure as well as temperature. Gases are far more compressible than liquids and solids. Over a normal temperature range many gases can be assumed to behave like ideal gases: Ideal Gases For ideal gases the density is directly proportional to the pressure: p · V = m · RS · T m p = V RS T p = RS T

(2.14) (2.15) (2.16)

so the density of the ideal gas is ∼p

(2.17)

The negative slope of the volume–pressure curve divided by the initial volume is called the compressibility . It is the inverse compression modulus K of a material. 1 dV =− · (2.18) V dp 1 =K 

(2.19)

Liquids and Solids Ideal liquids and solids show an elastic behavior. That means their volume can decrease by a certain amount when a pressure is applied, but that it will fully recover to the initial volume when the pressure is restored. For this type of material the change in volume on increasing the pressure can be calculated based on equation (2.20): V = − · p V with (2.8) and because of m = const the relative density is:     V  = | · p| =  V 

(2.20)

(2.21)

2.3 Density

where p Rs  T m V  K

49

pressure in Pa specific gas constant in J · K−1 · kg−1 density in kg · m−3 temperature in K mass in kg volume in m3 compressibility in Pa−1 modulus of compression in Pa

Liquids and solids with very low compressibility  show a very small volume reduction and often are treated in practice as incompressible materials. Water has very low compressibility with a value of  ≈ 5 · 10−10 Pa−1 . So, up to pressures in the order of 10 MPa (100 bar), the reduction of the volume in water is so small that it can be neglected.Although in high pressure processing of food, where the pressure will range to some 100 MPa, the compressibility of water cannot be neglected [2] (see Example 2.2), and also must be taken into account because the mechanical energy of compression is converted to heat (see Example 2.3 and Example 2.4). Example 2.2. Change in density of water under high pressure operations  = | · p|  With this equation the relative change in density is: p 10 bar 100 bar 1000 bar 10000 bar

 = 5 · 10−10 Pa −1 · 106 Pa = 5 · 10−4 =   = 5 · 10−10 Pa −1 · 107 Pa = 5 · 10−3 =   = 5 · 10−10 Pa −1 · 100 · 106 Pa = 5 · 10−2 = 

result 0.05% 0.5% 5.0%

 = 5 · 10−10 Pa −1 · 1000 · 106 Pa = 50 · 10−2 = 50% 

The modulus of compression can also be used to characterize the firmness or stiffness of materials (see Section 4.1.3). Example 2.3. Energy input by compression of 1 kg water by p = 300 MPa (3000 bar) dW = −p · dV  W = − p · dV

50

2 Mass and Density

 p · (− · V) · dp

W =−  W =

p · V · dp  W =  · V p · dp

1 W =  · V · p2 2 The dimensional units for this term are: N Pa−1 · m3 · Pa = 2 · m3 = N · m m 3000 bar = 300 MPa = 3 · 108 Pa sample: 1 kg Water  ≈ 5 · 10−10 Pa−1 W 1 V =  · · p2 m 2 m 2 J W 10−3  1 kJ = · 5 · 10−10 · 3 · 108 = 22.5 m 2 1 kg kg Example 2.4. How high will temperature rise in response to this energy input? Q = m · cp · T Q = cp · T m T =

Q m · cp

T = 22.5

kJ 1 kg · ·K kg 4.18 kJ

T = 5.4 K For calculation of temperature increase during high pressure processing of food refer also to reference [24]. 2.3.3 Speciˇc Gravity (Relative Density) The ratio of the absolute density of a material to the density of a reference material is called relative density d. Water at 4 ◦ C or 20 ◦ C is most often used

2.3 Density

51

as the reference material for this purpose. In the USA and Canada, when water is used as the reference standard, the term “relative density” is not used, and is replaced by the term “specific gravity.” Since water is nearly always chosen as the reference standard world-wide, for practical purposes the terms “relative density” and “specific gravity” may be considered as synonymous.  d= (2.22) R where d relative density (specific gravity)  density in kg · m−3 R density of reference material in kg · m−3 It is important to note that both density and specific gravity (relative density) relate to the same physical property. However, density must be reported in dimensional units of mass per unit volume (e.g. g · cm−3 or kg · m−3 ), while specific gravity (relative density) is a ratio of densities, and is always a dimensionless number. In the case where density is being reported in dimensional units of g · cm−3 and with the density of the reference material being 1 g · cm−3 or nearly 1 g · cm−3 it is interesting to note that numerical values of both density and specific gravity (relative density) will be the same, respectively, nearly the same (see Example 2.5). Example 2.5. Specific gravity of potatoes specific gravity d =

density of potatoes

with

 = 1000 kg · m−3

R = 1000 kg · m−3 (water of 4 ◦ C) d = 1.080

 R

R = 997.1 kg · m−3 (water of 20 ◦ C) d = 1.083

2.3.4 Methods for Laboratory Measurement of Density Table 2.4 shows examples of different methods used for density measurement. Pycnometric Measurement By weighing a known volume of a liquid, the density of that liquid can be measured in a simple way. Glass bulbs with precisely known volume that are used for this purpose are called pycnometers. A pycnometer can also be any other instrument designed for the same purpose that may have sample chambers of precisely known volume, but made of other materials (not glass bulbs). The glass bulb or sample chamber will have a marker to which the liquid sample must be carefully filled. Then the density of the fluid can be calculated by:

52

2 Mass and Density

Table 2.4. Techniques of density measurement name pycnometer

hydrostatic balance

hydrometer submersion balance for viscous liquids

density gradient column

for

fluids, solids

fluids, solids

fluids

fluids

Solid particles

Mohr– Westphal balance

X-ray technique

bulk density

tapped bulk density

resonator frequency

fluids

solids

solids

solids

gases, fluids

for

mF − m0 (2.23) V Because of thermal expansion of the glass, the pycometer volume is known for the temperature at which it was calibrated, only. So, for measurement of the absolute density,pycnometers should be used at the same temperature at which they were calibrated. Another way is to measure the relative density (specific gravity) rather than the absolute density. For this purpose, the pycnometer is weighed with the sample liquid and again weighed with the reference liquid (often water). The ratio of both weights gives the relative density d, or specific gravity of the sample. F =

mF mF · V F = = =d mW V · mW W

(2.24)

The advantage of the latter approach is that neither the weight of the empty pycnometer nor the volume needs to be known. So, if there is no opportunity to adjust temperature to accommodate pycnometer calibration, the instrument should be used to measure relative density (specific gravity). In this case, the weighing of the sample and weighing of the reference material must be carried out at the same temperature. m0 mF mW V F

mass of empty pycnometer in kg mass of pycnometer filled with sample in kg mass of pycnometer filled with water in kg volume in m3 density of sample in kg · m−3

2.3 Density

W d

53

water in kg · m−3

density of relative density of sample (specific gravity)

Once the relative density d, or specific gravity, of the sample is known, and the density of the reference material is known from the literature, the absolute density of the sample can be calculated: F = d · W

(2.25)

As mentioned in Section 2.2 a weighing result can be corrected for the effect of atmospheric buoyancy. The corrected mass is the true mass of a body, and is slightly higher than the mass which the balance displays. When a true mass is used to calculate density this is called a true density. If the atmospheric buoy-

Figure 2.4. Pycnometer designs: (a) Reischauer, (b) Bingham, (c) Gay-Lussac, (d) Sprengel, (e) Lipkin, (f) Hubbard [5,6]

Figure 2.5. Examples of pycnometers from [5]. The right one is for viscous samples and powders

54

2 Mass and Density

ancy correction is not applied, the result can be called an apparent mass, and respectively, an apparent density. When relative density is measured by using the same pycnometer for both sample and reference material, the buoyancy effect is eliminated by forming the weight ratio. Figure 2.4 and Figure 2.5 show different designs of glass pycnometers. Hydrostatic Balances (Buoyancy Weighing) The principle of a hydrostatic balance is based on Archimedes law of buoyancy. If a body is submersed in a fluid its weight will be lowered because of the buoyancy force. The buoyancy force is directly proportional to the volume of the submersed body and the density of the fluid. By measurement of the buoyancy force with the balance, the volume of the body can be determined quite accurately, and together with the measured mass of the body, the density is obtained. A simple technique for making this type of measurement is to place a beaker partially filled with water on top of a top-loading balance, with the weight of beaker and water tared-out to read zero on the display. Then, fully submerge the solid body beneath the water surface, taking care that it neither touches the bottom nor the sides of the beaker. The weight reading shown on the display of the balance will be the weight of the volume of water displaced by the solid body. Since density of water is known, the precise volume of the solid body is determined. mL K = (2.8) VK FA = F · VK · g

(2.26)

with m as the difference in weight (in kg) of the body before and after submersion: m = mL − mF

(2.27)

FA = m · g

(2.28)

(mL − mF )g mL − mF = F · g F mL · F K = mL − mF VK =

where mL mF FA g VK F K d

mass of the body in air in kg mass of the body submersed in fluid in kg buoyancy force in N gravitational acceleration in m · s−2 volume of body in m3 density of fluid in kg · m−3 density of body in kg · m−3 relative density

(2.29) (2.30)

2.3 Density

55

So the density of a body can be obtained by taking first the weight prior to submersion (that means weighing in air) and its weight when submersed in a fluid with a known density F using equation (2.30). With the ratio K mL = F mL − mF

(2.31)

the relative density d (specific gravity) of the body is K d= F or mL d= mL − mF

(2.32)

(2.33)

That means d can be calculated very quickly after two readings from the balance. If the weight in air mL is not corrected for atmospheric buoyancy, the density obtained by hydrostatic weighing can be called apparent density of the body. If that correction made, then mL and K will be slightly higher, and can be called true mass and true density. On the other hand when a body of known volume is submersed in a fluid, the difference in weight of the body in air and the weight of fluid displaced by the body can be used to determine the density of the fluid, and can be calculated with equation (2.34): mL − mF F = (2.34) VK Figure 2.6 shows a special design of hydrostatic balance that is suitable for measuring the density of a solid or the density of the liquid in the reservoir when used with a solid body of precisely known volume. To obtain the density

Figure 2.6. Hydrostatic balance design. 1: balance, 2: platform, 3: small beaker, 4: large beaker, 5: support bracket, 6: pan, 7: thermometer

56

2 Mass and Density

of a solid, the sample is weighed first in air, and then it is submersed and the weight is taken again.As can be seen in Figure 2.6 there is a small pan mounted on the weighing plate of the balance. A small beaker on a cable is suspended within a larger beaker containing the fluid of interest. The large beaker is resting on a raised platform so its weight is not transmitted to the balance. To obtain the fluid density in the large beaker, a test body with known volume is first placed on the pan and its weight in air is measured. Then the test body is placed into the small beaker, submersed in the fluid and weighed once again. Then the density of the liquid is calculated using equation (2.34). It should be remembered that the density of the fluid is dependent on temperature so the temperature must be controlled and recorded. Here are two examples of application the principle of a hydrostatic balance. Example 2.6. Starch in corn kernels Agronomists have determined that degree of ripeness of an ear of corn can be closely correlated with density (or specific gravity) of individual corn kernels. A specific gravity (relative density) between 1.080 and 1.118 is the range expected at peak ripeness. Values below that range would indicate insufficient ripeness, while values above would indicate the corn has gone beyond peak ripeness [110]. With a hydrostatic balance a fast check of incoming raw material is possible. Example 2.7. Starch in potatoes Another application is in checking the quality of incoming potatoes for a potato processing facility. The density of potatoes also is a function of the starch content. By weighing a basket with potatoes in air, and then submersed in water, the starch content can be estimated quickly. The diagram in Figure 2.7 shows the relationship between the relative density (specific gravity) of potatoes and their starch content. For convenience, the scale on the left side vertical axis of the diagram shows the underwater weight mUW of a potato sample weighing 5050 g in air. A standard test with that sample size is published by the EU [3]. The fact that the weight of the basket in air and under water is not the same is neglected. The underwater weight force of the sample is the weight force in air GL lowered by the buoyancy force FA :

Figure 2.7. Relative density d of potatoes versus starch content. The scale on the left shows the underwater weight of a 5050 g sample

2.3 Density

57

GUW = GL − FA

(2.35)

Hence mUW · g = mL · g − F · VK · g so the volume of the sample is mL − mUW VK = F

(2.37)

and the density of the sample is mL F · mL K = = VK (mL − mUW ) The relative density then is mL K = d= F mL − mUW where g GUW GL FA mUW mL F Vk d

(2.36)

(2.38)

(2.39)

gravitational acceleration in m · s−2 underwater weight force in N weight force in air in N buoyancy force in N apparent mass of submersed sample in kg mass of sample in air in kg density of fluid in kg · m−3 volume of sample in m3 relative density (specific gravity) of sample

Mohr{Westphal Balance The Mohr–Westphal balance (see Figure 2.8) is another type of hydrostatic balance. It is designed as a nonsymmetric beam balance for measuring the density of liquids.

Figure 2.8. Mohr–Westphal balance. 1: beam, 2: weights, 3: buoyancy body, 4: liquid sample

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2 Mass and Density

At the free end of the arm of the balance a “buoyancy body” is suspended in air. The buoyancy body is normally made of glass and can have a built-in thermometer. Then the buoyancy body is submersed into the liquid of interest. Because of the effect of buoyancy, the weight of the submersed glass body will appear lower than it was in air, and will bring the balance out of zero. The buoyancy force can be measured by successively adding small weights to the arm until the balance is restored to zero. The measurement is then repeated with water as a reference liquid. The ratio of both readings provides the relative density or specific gravity of the liquid as can be shown below. With the buoyancy force FA =  · V · g

(2.40)

FA = m · g

(2.41)

with the fluid of interest FA,F = F · VK · g = mF · g

(2.42)

with water as reference material FA,W = W · VK · g = mW · g so the ratio is mF = mW

(2.43)

(2.44)

Because the volume of the glass body is the same for both readings, then =

mF mF · V F = = =d mW V · mW W

so the specific gravity is simply mF d= mW where m V FA  g  d subscript F subscript W

(2.45)

(2.46)

mass of weights in kg volume in m3 buoyancy force in N density in kg · m−3 gravitational acceleration in m · s−2 submersed weights ratio relative density (specific gravity) for fluid for reference material (mostly water)

The result should be recorded along with the temperature of the measurement. Often both readings are taken at 20 ◦ C and the result is written as d20/20. The quantity d20/4 would mean that the density of the liquid was compared to the density of water at 4 ◦ C:

2.3 Density

d20/20

59

F,20 ◦ C = W,20 ◦ C

d20/4 =

F,20 ◦ C W,4 ◦ C

The Mohr–Westphal balance can also be used to measure the density of a solid sample which is submersed in a fluid with known density. For this purpose, equation (2.37) would be used to calculate the volume of the sample. Hydrometer Hydrometers (sometimes called areometers) are hollow glass bodies with the shape of a buoy. Hydrometers are designed with a volume to mass ratio in such a way that the glass body will float at a certain depth in the liquid under investigation. Depending upon the density of that liquid the hydrometer will float at a higher or lower position. The upper part of the hydrometer has a scale for reading the nonsubmersed part h of the floating glass body. The floating depth position of the hydrometer depends on weight force and interfacial force (force due to surface tension, see Section 5.1). It is FG + F = FA

(2.47)

which means: so

m · g +  ·  · d = VS ·  · g

(2.48)

   · d2 ·h · ·g m·g + · ·d = V − 4

(2.49)

The nonsubmersed part of the hydrometer has the length   m·g + · ·d 4 V− h=  · d2  ·g

(2.50)

which means h = function ()

(2.51)

The nonsubmersed length of the hydrometer can be read with the aid of a scale on the upper part of the hydrometer.A weight at the bottom of the hydrometer acts like the keel of a sailboat to ensure that it will float in the liquid in a vertical orientation. The scale can be calibrated directly in units of density or, e.g. in concentration units (see Figure 2.10). Hydrometers with special scales are available for specific applications, such as for sugar solutions (saccharimeter), alcohols (alcoholometer), acids (acid hydrometer), Baum´e hydrometer for salt solutions, milk (Quevenne lactometer), beer wort hydrometer, urine checker, liquid gas aerometer, etc.

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2 Mass and Density

Figure 2.9. Hydrometer. 1: scale, 2: body (with and without thermometer), 3: keel

Figure 2.10. Reading of a hydrometer scale at the liquid surface (example)

Sometimes the combination of two physical properties will give the information needed about a process or a product. Forexample, by knowing both the density and refractive index of beer wort, the alcohol content can be calculated, and by this, the progress of fermentation [101,111]. m V VS FG FA F  d  g h

mass of hydrometer in kg total volume of hydrometer in m3 volume of hydrometer submersed in m3 weight force of hydrometer in N buoyancy force in N interfacial tension force (force of surface tension) in N interfacial tension (surface tension) in N · m−1 diameter of hydrometer neck in m density of liquid kg · m−3 gravitational acceleration in m · s−2 length of hydrometer neck not submersed in m

Without consideration of interfacial tension effects the hydrometer equation simplifies to   m 4 (2.52) V− h=  · d2  There are hydrometers available which are corrected for interfacial tension offered in different range categories called L (low, 15–35 mN · m−1 ), M (medium, 35–65 mN · m−1 ) and H (high, for higher values).

2.3 Density

61

Submersion Technique Pycnometers and hydrometers do not work very well with liquids of high viscosity. For highly viscous liquids, measurement of density can be performed with the submersion technique (see Table 2.4).A beaker with the viscous liquid sample is put on a balance. The display value is recorded, or the display may be set to zero (tare). Then a test body with known volume is pressed into the sample.The buoyancy force caused by the submerged test body is transferred to the balance and appears on the display as an apparent increased weight m. This increased weight force is the buoyancy force, and is the weight of the displaced liquid, which is equal in volume to the volume of the submersed solid body: G = FA

(2.53)

and m · g =  · VK · g

(2.54)

so = where G m VK 

m VK

(2.55)

apparent weight force increase in kg apparent mass increase in kg volume of test body in m3 density of liquid kg · m−3

For precision measurement, the buoyancy body can be a hollow metal sphere with calibrated volume. To avoid errors from buoyancy of the mounting rod there is normally a depth mark on the rod which indicates the right depth position for immersion so that the submerged section of rod is accounted for in the calibrated volume.

Figure 2.11. Submersion technique for density measurement. 1: depth mark, 2: liquid sample, 3: buoyancy body of known volume

62

2 Mass and Density

Particle Floating Technique A solid particle will remain suspended within a liquid (meaning it will neither rise to float at the surface nor sink to rest at the bottom) when both the liquid and solid particle have the same density. Then the buoyancy force acting upward on the particle will be equal to the gravitational force acting downward, and the particle will remain suspended. To measure the density by the floating method, two different liquids must be chosen so that one has a density lower than the particle, and the other higher.Also both liquids must be miscible with each other when mixed together, and remain optically transparent in order to observe the particle when placed into the liquids. The liquids must also have no adverse effect on the particle, meaning the particle will not dissolve or be altered in any way. Appropriate liquids for this purpose are listed in Table 2.5. Table 2.5. Liquids for the floating technique (examples) liquid water aqueous sucrose solution 50% (m/m) aqueous NaCl solution 20% (m/m) ethanol 90% (m/m)

density at 20 ◦ C in kg · m−3 [137] 998.2 1229.5 1147.8 818.0

The solid particle of unknown density is first placed into a beaker partially filled with the low density liquid. Since the liquid density is lower than that of the particle, the particle should sink to rest at the bottom of the beaker. Then, small amounts of the high-density liquid are lowly added and gently mixed until the particle just begins to float upward. When this point is reached the density of the liquid mixture and the solid sample are the same (Figure 2.12). By measuring the density of the liquid mixture, e.g. by a pycnometric method, the density of the solid particle sample is known. For measuring the density of agricultural materials or food products, the floating method liquids will need to have densities ranging between 900 and 1500 kg · m−3 . This range of densities can be obtained e.g. with mixtures of water and alcohols or with sugar solutions or salt solutions. The temperature of the liquid mixture must be held constant for this procedure to yield accurate results. If this is ensured there is no need to perform the floating technique at a fixed or controlled temperature so it can be carried out very easily. The floating technique is recommended for solid samples which cannot be handled in a hydrostatic balance. Problems may arise because the sample may dissolve or alter in aqueous solutions. In that cases the samples should be packaged or coated or another liquid should be chosen.

2.3 Density

Figure 2.12. Floating technique. The sample P is suspended in the fluid F

63

Figure 2.13. Density gradient column (from [133]). 1: clock motor, 2: water jacket, 3: elevating basket, 4: stand

Density Gradient Column An interesting variation of the floating technique is the use of a density gradient column as a fast and accurate method for determining the density of small solid particles like agricultural seeds and grains. The method is based on observing the depth to which a particle will remain suspended in a tall column of liquid that exhibits a gradual change in density along the depth of the column, as shown in Figure 2.13. The column is first prepared by filling it carefully with layers of mixtures in different proportions of two miscible transparent liquids of different densities similar to those used in the floating methods. The first layer at the bottom would consist only of the high density liquid. The next layer above would be a mixture of 10% low density liquid and 90% high density liquid, followed by a layer of 20% low density with 80% high density, and so on, until the layer at the top would consist only of the low density liquid. In this way, the column will exhibit a gradual gradient in density ranging from the high density at the bottom up to the low density at the top. The column is normally calibrated with color-coded glass beads of known density commercially available for this purpose. The calibration beads are dropped into the column, and each bead comes to rest suspended at the depth to which the liquid density is the same as the bead. A graph of bead density versus column depth serves to calibrate the column. Then, small particles of unknown density are dropped into the column. They will come to rest suspended at the depth to which the liquid density is the same as the particle. This depth is read from the gradu-

64

2 Mass and Density

Table 2.6. Liquids suitable for density gradient column (from [133]) liquid isopropanol–water isopropanol–diethylene glycol water–calcium nitrate

specific gravity 0.79 to 1.00 0.79 to 1.11 1.00 to 1.60

ated scale on the column. The particle density is obtained from the calibration graph, which will give the density at that depth. Table 2.6 gives a listing of various liquid mixtures suitable for preparation of a density gradient column. Resonator Frequency Techniques When we have an oscillating tube filled with the fluid of interest, its resonant frequency depends on the density of the liquid. So with electronic means and appropriate calibration, it is possible to measure the density of a fluid by monitoring the resonant frequency of a oscillating (vibrating) tube. Density meters based on this effect are available as stationary laboratory devices, portable hand-held sensors and as in-line process sensors. The working principle is described in detail in Section 14.3.1. Overrun Overrun is a term used in the manufacture of products involving whipping or aeration, such as in ice cream or whipped dairy toppings. The whipping process causes air to become incorporated into the liquid being mixed causing the physical phenomenon of foaming. The density is an important process parameter which is directly related to the degree foaming. Foaming causes the volume of the foamed liquid to increase far beyond the original volume of the liquid prior to whipping. This volume increase, when divided by the original volume, is called overrun A. It is a dimensionless ratio, and can be mathematically expressed as follows: A=

Vfoam − Vliquid Vliquid

(2.56)

since: mfoam = mliquid + mgas ≈ mliquid

(2.57)

and with m = mfoam ≈ mliquid with V 1 =  m

(2.58)

2.3 Density

65

the overrun can be expressed: m m − Vfoam Vliquid A= m Vliquid

(2.59)

i.e. 1 A=

foam

−

1 liquid

1

(2.60)

liquid which means liquid liquid − A= foam liquid so liquid −1 A= foam

(2.61)

(2.62)

Equation (2.62) yields a dimensionless ratio. The equation was developed from (2.56) which indicates that the ratio is a volume ratio. So when the overrun is calculated in %, the meaning is a percentage of volume.   liquid A in % = − 1 · 100 (2.63) foam where A m V 

overrun mass in kg volume in m3 density in kg · m−3

Example 2.8. Overrun of ice cream Ice cream is produced by whipping and freezing of a liquid mixture (see e.g. [109]). Typical values are given to ice cream = 550 kg · m−3 and ice mix = 1150 kg · m−3 . The overrun is calculated using equation (2.62) to   liquid − 1 · 100 % A= foam   1150 kg · m−3 A= − 1 · 100 % = 109 % 550 kg · m−3 This value for the overrun means the volume of the liquid ice cream mix was raised by 109%, or more than doubled.

66

2 Mass and Density

Density of Solids Many agricultural materials and food and feed ingredients are in the form of granular materials (grains, meals and powders), which are bulk solids made up of small particles. The weight or size of the individual particles within any of these types of materials may vary over a large range e.g. from frozen diced vegetables to corn cornels to fine powder particles. The term “solid density” means the density of the solid material of which a particle is made, no matter what type of fluid or other material may exist between the particles. On the other hand if the solid particles contain pores or hollow cavities filled with gases or liquids, this contributes to the density of the solid. When pores or cavities occur, it is important to state whether they are closed or open. If they are closed, meaning they are located completely within the solid particle, they belong to the solid. If they are open to the surroundings at the particle surface, e.g. the atmosphere, they do not belong to the solid body. To avoid communication errors the density of solids should be given with a note like “including pore volume” or “without pore volume.” This can be accomplished by how the system boundaries are defined, and can be calculated as follows: mS = (2.8) VS where mS mass of solid material in kg VS volume of solid material in m3  density of solid in kg · m−3 To measure the density of a solid particle often simply means to measure its volume, because its mass is known upon weighing. To get the volume of the solid sample without its open pores, a pycnometer technique with an appropriate liquid can be used. The liquid must not alter or dissolve the sample. For this purpose, a sequence of weighings is conducted as indicated in Figure 2.14. After weighing the empty pycnometer m0 the pycnometer is weighed with the sample mP . Then the pycnometer is filled up to a designated mark with an appropriate reference liquid of known density and weighed again mP,F . Finally, the pycnometer is weighed when filled to the same mark with only the reference liquid mF [4]:

Figure 2.14. Pycnometer measurement of the density of a solid granular material, e.g. a powder

2.3 Density

67

The density of the solid material then is mS S = VS

(2.8)

mS = mP − m0

(2.64)

VS = V0 − VF mP,F − mP VF = F

(2.65)

S =

mP − m0 mP,F − mP V0 − F

mP − m0 S = m mP,F − mP F − F F S = F

mP − m0 mF − mP,F + mP

(2.66) (2.67)

(2.68)

(2.69)

d=

S F

(2.70)

d=

mP − m0 mF − mP,F + mP

(2.71)

where m0 mS mP mP,F mF V0 VS VF S F

mass of empty pycnometer in kg mass of sample in kg mass of pycnometer with sample in kg mass of pycnometer with sample and filled with reference liquid in kg mass of pycnometer with reference liquid only in kg volume of pycnometer in m3 volume of sample in m3 volume of reference liquid filled in pycnometer in m3 density of solid in kg · m−3 density of reference liquid in kg · m−3

For precision measurements the true mass instead of apparent mass (see Section 2.1) should be used. In that case the true density of the reference liquid should also be used. Bulk Density Powders and bulk goods contain hollow spaces or voids filled with gas, normally air. The density of that type of bulk material, including the void spaces, is called bulk density. Using equation (2.8) the bulk density can be calculated by weighing a sample of the bulk material and measurement of its volume.

68

2 Mass and Density

The volume of the whole bulk material must be taken “as is.” To measure this volume, the sample material can be poured into a beaker or cylinder up to a known volumetric mark. Different technique in filling the beaker or cylinder may lead to different distributions of solid particles and hollow spaces. So, to get repeatable results the technique of filling has to be standardized. To overcome problems with repeatable filling technique, the bulk material can be tapped before reading of the volume. By tapping the material, the solid particles will “settle” into the most stable situation they can reach. The void spaces will get smaller as the solid particles settle step by step into a spatial situation where the bulk density will reach a maximum. The time needed to reach this maximum depends on tapping speed and tapping amplitude. There are laboratory devices available which perform a chosen number of tappings in a continuous motion. Depending on the sample properties, sometimes 10, 100, 1000 or more tappings may be appropriate to reach maximum bulk density. Figure 2.15 shows an example of a device which can be used to measure bulk density and tapped bulk density subsequently. First the bulk material is filled into a 1000 cm3 cylinder until it is overflowing under repeatable technique conditions. Then with aid of a flat spatula, the excess overflow of sample material is scraped away from the top of the cylinder to leave the sample perfectly level at the top, and the 1000 cm3 sample is weighed to get the bulk density. Now a cylindrical extension overring is slipped onto the 1000 cm3 cylinder and more sample material is filled in. The cylinder is mounted on the tapping device and moved for a fixed number of tappings. In German testing standards (DIN EN 1237 in [101]) a number of 2500 with a frequency of 250 s−1 is specified.

Figure 2.15. Device for tapping of bulk goods: 1: rotating cam, 2: housing, 3: powder sample, 4: cylinder, 5: overring

2.3 Density

69

Table 2.7. Characterization of powder flowability by Hausner ratio [141] Hausner ratio 1.0–1.1 1.1–1.25 1.25–1.4 > 1.4

powder flowability free flowing medium flowing difficult flowing very difficult or nonflowing

After this the sample material is adjusted to 1000 cm3 again and weighed. The tapped bulk density should be recorded with the parameters of its measurement. The difference between bulk density and maximum tapped bulk density provides information about the ability of the bulk material to be compressed by gravity or pressure. Powders can be characterized for this property by the Hausnerratio, which is the quotient of tapped bulk density over untapped bulk density (see Table 2.7). Porosity Also the volume of the hollow void space (pores) can be calculated. The ratio of the volume of the void space (pores) and the total volume of the bulk is called porosity ": VH "= (2.72) VB "=

VS V B − VS = 1− VB VB

because mB ≈ mS = m VS · m " =1− m · VB so B " =1− S

(2.73)

(2.74)

(2.75)

With " as the relative volume of the hollow pore space, and ˛ as the relative volume of the solid particle space it is evident that: ˛+" =1 where VB VH VS B S mB mS

total volume bulk in m3 volume of hollow space in bulk in m3 volume of solid material in bulk in m3 density of bulk in kg · m−3 density of solid in kg · m−3 mass of bulk in kg mass of solid in kg

(2.76)

70

2 Mass and Density

" ˛

porosity of bulk relative volume of solids in bulk

The porosity of an unconsolidated mass of bulk agricultural materials such as silage, straw and hay is a very important physical property that is needed in air flow and heat flow processes, as well as other applications. Measuring the solid volume of the particles in these types of materials can be very difficult.They are not suitable for being in contact with liquids used with pycnometer methods. For these types of materials, their natural porosity can determined directly with the use of a simple apparatus called porosity tanks. In this apparatus, two tanks of equal volume that can be closed air-tight are connected by a manifold of tubing with shut-off valves and a manometer as shown in Figure 2.16.

Figure 2.16. Porosity tanks (from [133])

Tank 2 is filled with sample material and sealed. Valves 2 and 3 are closed, and compressed air is brought into tank 1. When a suitable manometer displacement is achieved, valve 1 is also closed, and the pressure in tank 1 p1 is read and recorded from the manometer. Then, valve 2 is quickly opened, and the new lower equilibrium pressure in the system p3 is measured and recorded from the manometer. From the perfect gas law at constant temperature, the mass of air added to the system when tank 1 was initially pressurized m can be expressed as: m=

p 1 · V1 RS · T 1

(2.77)

After valve 2 is opened, this same mass becomes distributed between the empty volume of tank 1 m1 and the available hollow pore spaces in tank 2 m2 containing the sample. Then, by conservation of mass: m = m1 + m2

(2.78)

p 1 · V1 p 3 · V1 p 3 · V2 = + RS · T 1 RS · T 1 RS · T 1

(2.79)

and "=

V2 p 1 − p 3 = V1 p3

(2.80)

Literature

where " p V m T RS

71

porosity of bulk pressure in Pa volume in m3 mass in kg temperature in K specific gas constant in J · K−1 · kg−1

2.4 Applications alcohol concentration of drinks, pycnometer alcohol concentration of drinks, hydrometer solids in sugar syrups solids in milk milk, density by hydrometer relative density of fruit and vegetable juices relative density of beer wort relative density of fruit and vegetable juices animal and vegetable fats and oils: determination of conventional mass per volume (“liter weight in air”) specific gravity of water and brine density and specific gravity of viscous materials by Bingham pycnometer density and specific gravity of viscous materials by Lipkin bicapillary pycnometer density, specific gravity and absorption of fine aggregate density, specific gravity and absorption of coarse aggregate specific gravity of soil solids by water pycnometer specific gravity, apparent, of liquid industrial chemicals specific gravity of pigments

[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

Literature 1. CIPM, Comite Internationale des Poid et Mesures (1979) PTB-Mitt 89:271–280 2. Ludwig H (ed) (1999) Advances in High Pressure Bioscience and Biotechnology, Springer, Berlin 3. EU comission (ed) (1995) EG-Verordnung No 97/95 of Jan 17 1995,Amtsblatt Jan 24 1995, available online http://europa.eu.int/eur-lex/hu/dd/docs/1999/31999R2718-HU.doc 4. DIN ISO 8130-3 (2005) Determination of density by liquid displacement pycnometer, in [101] 5. DIN ISO 3507 (2002) Laboratory glassware – Pycnometers, in [101] 6. DIN 12807 (1975) Laboratory glassware, Bingham pycnometer, in [101] 7. AOAC method 930.17, in [104] 8. AOAC method 9503, in [104]

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2 Mass and Density

9. AOAC method 932.14 B, in [104] 10. AOAC method 925.22, in [104] 11. LFGB method L01.00-28 (1988) Untersuchung von Lebensmitteln; Ar¨aometrische Bestimmung der Dichte von Milch, in [100] 12. LFGB method L31.00-1 (1997) Bestimmung der relativen Dichte von Frucht- und Gem¨uses¨aften, in [100] 13. LFGB method L36.00-3 (1994) Untersuchung von Lebensmitteln; Bestimmung der relativen Dichte d 20/20 von W¨urze und Bier, in [100] 14. DIN EN 1131 (1994) Fruit and vegetable juices – Determination of the relative density; in [101] 15. ISO 6883 (2006) Animal and vegetable fats and oils – Determination of conventional mass per volume (‘litre weight in air’), in [101] 16. ASTM D 1429-03 Standard Test Methods for Specific Gravity of Water and Brine, in [132] 17. ASTM D 1480-02e Standard Test Method for Density and Relative Density (Specific Gravity) of Viscous Materials by Bingham Pycnometer, in [132] 18. ASTM D 1481-02 Standard Test Method for Density and Relative Density (Specific Gravity) of Viscous Materials by Lipkin Bicapillary Pycnometer, in [132] 19. ASTM C128-04a Standard Test Method for Density, Relative Density (Specific Gravity), and Absorption of Fine Aggregate, in [132] 20. ASTM C127-04 Standard Test Method for Density, Relative Density (Specific Gravity), and Absorption of Coarse Aggregate, in [132] 21. ASTM D854-06 Standard Test Methods for Specific Gravity of Soil Solids by Water Pycnometer, in [132] 22. ASTM D891-95 (2004) Standard Test Methods for Specific Gravity, Apparent, of Liquid Industrial Chemicals, in [132] 23. ASTM D153-84 (2003) Standard Test Methods for Specific Gravity of Pigments, in [132] 24. Patazca E, Koutchma T, Balasubramaniam VM (2007) Quasi-adiabatic temperature increase during high pressure processing of selected foods.J Food Engineering 80:199–205

3 Geometric Properties: Size and Shape

Geometric properties are those that can be derived from the geometry of a solid body or particle. They are very important as a means by which the size and shape of an irregular shaped particle can be easily quantified. This is true whether the particle is treated as an individual solid body, or as one that is representative of many particles in a disperse system. Material systems that consist of particles surrounded by a continuous medium are known as disperse systems. The particles form the disperse phase, and the surrounding medium is the continuous phase. The size of the particles can vary from the magnitude of 10−1 m (grapefruit, melons, potatoes) to 10−6 m (emulsions), and down to 10−8 m (colloids and nanoparticles). Particles can be solid, liquid or gaseous. Six different classes of disperse systems are known (Table 3.1). Table 3.1. Classes of disperse systems disperse phase solid

continuous phase gaseous

disperse system

example starch powder, soy beans, corn kernels

liquid

powders, dust, bulk good suspension

solid liquid liquid gaseous

liquid gaseous liquid

emulsion aerosol foam

gaseous

solid

solid foam

aqueous starch suspension, ketchup, cocoa drink, liquid chocolate (molten) mayonnaise, milk, salad dressing fog/mist, spray coatings whipped cream, ice cream, beaten egg whites marshmallows, bread, angel food cake

Disperse systems can be characterized by specifying the size and form (or shape) of their particles. For example let us imagine a disperse system consisting of sugar particles (crystals) in air (a bowl of sugar). Depending on size and shape of the sugar particles we can distinguish numerous different products with different physical properties like porosity, bulk density, and flow properties (Table 3.2). In Figure 3.1 an overview is given on magnitudes of particles sizes which occur in food engineering. In this illustration it can be seen that the size of

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3 Geometric Properties: Size and Shape

Table 3.2. Sucrose in air: examples in the class of solid–gaseous disperse systems English Rock candy sugar crystals Dark brown sugar Light brown sugar Refined (white) table sugar Confectioners’ sugar (powdered sugar) Cotton candy

German Kandiszucker Brauner Zucker Teezucker Haushaltszucker Puderzucker Zuckerwatte

Figure 3.1. Magnitudes of particle sizes and comparison to other lengths

particles, microorganisms and molecules can be compared to the wavelength of electromagnetic radiation. Particle size and shape are of decisive influence in governing the physical properties of the disperse system. For example, flow properties (see Section 4.3), bulk density, porosity, stability of foams and emulsions depend on it. The interfacial surface area of the bulk materials (sometimes referred to as the “specific surface area”) can be thought of as the total surface area exposed to the hollow spaces in between the particles in the disperse system. The specific surface area will depend on particle size and shape, and will affect the kinetics of sorption, dissolution or reactions that may take place involving the disperse system. For a general definition of particle size there is no need to specify the various classes of disperse systems or to make any distinction between solid particles, liquid droplets or gaseous bubbles. On the other hand, especially in

3.1 Particle Size

75

the case of nonspherical particles, the method or technique by which particle size is measured may influence the end result. Let us imagine an irregular shaped particle like a kidney bean. If we were to attempt to draw its shape on a plane surface like a sheet of drawing paper, it is evident that the particle shape and size we draw will depend on the direction from which we observe the particle. This same situation is true when trying to describe the shape of the many irregular-shaped bodies of materials found among food and agricultural products. Figure 3.2 shows examples of charts prepared in an attempt to describe the shapes of apples, peaches and potatoes in some type of standard way. In addition, not all the beans in a box of kidney beans, nor all the apples in an apple bin, nor all the peaches in a basket of peaches, nor all the potatoes in a sack of potatoes will have the same shape and size. This is the reason why particle size and shape must be dealt with in terms of mathematical distributions. This chapter is about measurement of particle size and shape, as well as particle size and shape distributions in disperse systems, and how these distributions can be quantified and understood. First we will focus on different definitions for the size of an individual particle. Particle size distributions will be treated in Section 3.2.

3.1 Particle Size Particles with a regular shape like cubes, rectangular blocks, cylinders or spheres can be characterized by their linear dimensions (lengths) along their principal axes. For example, the size of a cube can be given by specifying the length of any one side,the size of a rectangular block can be given by specifying its three orthogonal (mutually perpendicular) dimensions (length, width and height), the size of a cylinder by its length and diameter, and the size of a sphere by its diameter. Also, the volume and surface area of such solid bodies with regular geometries can be easily calculated by well known mathematical equations that will give volume and surface area as a function of these basic linear dimensions. Normally, a Cartesian system of coordinates is used for this purpose, and one axis is identified as the symmetric axis of the particle. Another way to characterize particle size is to use the volume or the surface area of the particle. So-called equivalent particle sizes are hypothetical dimensions that could be assigned to a model particle with the same volume or surface area. For example, it is often common practice to assign an “equivalent diameter” to an irregular-shaped particle. This would be the diameter of a perfect sphere having the precise same property, e.g. the same volume as the irregular shaped particle (see Section 3.1.2).

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3 Geometric Properties: Size and Shape

Figure 3.2. Example of charted standards for describing shape of fruits and vegetables (from Mohsenin [133])

3.1 Particle Size

77

3.1.1 Sizing by Image Analysis When we look at a particle through a microscope we can only see the projection (“shadow“) of the particle on a two-dimensional flat plane that is perpendicular to the direction of view through the microscope. When we place a micrometer scale near the particle on the microscope, we can estimate the size of the particle in only two mutually perpendicular dimensions on the same plane (directions x and y). In that case the directions x and y are perpendicular to each other on the same plane, and could represent the length and width of the particle. But, we would not know the height or thickness, which would be the third dimension in the direction of observation z. Therefore, both directions (x and y) are mutually perpendicular to the direction of observation z. In order to measure the particle thickness in the z-direction, the particle would have to be rotated by 90◦ so that the two-dimensional plane that could be seen would contain the mutually perpendicular directions of (x and z or y and z).Figure 3.3 illustrates these main particle diameters with the example of a kidney bean. Figure 3.4 gives support in understanding how a particle has a maximum area of projection (upper picture), a minimum area of projection (lower picture) and a case in between.

Figure 3.3. Major diameter a, minor diameter c and intermediate diameter b of a particle (kidney bean) can be identified from different projections of the particle (see also Figure 3.4).

Figure 3.4. Different projection areas of a parallelepipedon allow us to identify the major diameter a, minor diameter c and intermediate diameter b

An average of these three mutually perpendicular dimensions of length a, width b and thickness c, as defined here can often be used to specify the average size of the particle using a single characteristic diameter. This can be 1 3 an arithmetic mean diameter a+b+c 3 ,or a geometric mean diameter (a·b·c) .To avoid basic errors, the length to be measured or used to calculate sizes should be specified carefully. Table 3.3 shows some examples of other characteristic

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3 Geometric Properties: Size and Shape

Table 3.3. Some characteristic diameters of particles (examples)

Feret diameter Martin diameter major diameter

definition from Figure 3.5 height of projected area (top to bottom) diameter which cuts projected area into equal area parts (but may be different shape) diameter of smallest circumference

symbol xFe xMA a

Figure 3.5. Characteristic length of particles (examples). The Feret diameter gives overall height from top to bottom. The Martin diameter cuts the projected area into equal area parts. The major diameter a is the very longest diameter completely inside the projected area stretching across opposite ends of the projected area

diameters, which can be used to express particle size, and Figure 3.5 gives the definition of these diameters. Investigating a number of particles will show that whatever characteristic diameter is used to specify the size of a particle, the actual size of the individual particles will vary to a certain extent within the population of particles. Only in an ideal or theoretical situation will particles all have the same diameters.Many real particles in nature, such as soy beans and oranges, can often be assumed to be characterized by an ideal shape, such as a perfect sphere. Whether or not the error which is introduced by this assumption is significant depends on the application and precision needed to solve the engineering problem of interest. To have greater certainty about the true dimensions and shapes of the particles within a collection of particles a large number of particles should be examined and measured and quantified with the use of appropriate statistic procedures. 3.1.2 Equivalent Diameters An equivalent diameter of an irregular shaped particle is the diameter of a hypothetical ideal geometric body, such as a perfect sphere, having the same specified properties as the real particle. Depending upon on these specified properties, e.g. geometric property or physical property, there are different definitions for equivalent diameter. Some of them will be introduced in the following subsections.

3.1 Particle Size

79

3.1.3 Geometric Equivalent Diameters For any irregular shaped particle there exists an hypothetical spherical particle which has the same volume. The diameter of this hypothetical sphere is an equivalent diameter for the real particle and called the diameter of volumeequivalent sphere. This is perhaps the most frequently used type of equivalent diameter for most engineering applications. In the same manner alternative equivalent diameters may be chosen to be the diameter of sphere having the surface area as the real particle, or the diameter of a perfect circle enclosing the area as the projected area of the real particle, or diameter of a perfect circle having a circumference equal to the perimeter of the projected area of the real particle. Table 3.4 gives an overview of these geometric equivalent diameters. Table 3.4. Geometric equivalent diameters diameter of volume equivalent sphere

surface area equivalent sphere

projected area equivalent circle

perimeter circumference equivalent circle

formula  6·V dV = 3   A dA =   4·S dP =  dPe =

U 

to be measured is the: volume V

surface area A

projected area S

perimeter U of projection area

A particle like a cube or a sphere has the same diameter independent of the direction of observation or direction of measurement. This class of particles is said to have a isotropic shape. “Isotropic” means “being the same in all spatial directions.” If particles are anisotropic (nonisotropic) like bricks, cylinders or needles, an additional problem of experimental measurement arises: For example, if they are suspended in a liquid they will be oriented randomly, so measurement of a large number of particles will yield a truly representative average of their dimensions over all axes. However, if the same type of particles were sprinkled on a flat surface or plate for observation with a microscope or camera, their orientation would not be completely randomized. Needles for example would all tend to lie down on the flat surface, and would not stand upright. So, in case of particles with anisotropic shape, the technique of measurement should be chosen carefully and accurately reported or documented to avoid the possibility that observations and measurements from image analysis could be misinterpreted.

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3 Geometric Properties: Size and Shape

3.1.4 Physical Equivalent Diameters For any irregular shaped particle there also exists a hypothetical spherical particle which has the same physical behavior in a given process situation, such as having the same terminal velocity of free fall in a given fluid (velocity of sedimentation in a liquid, see also falling sphere viscosimeter, p. 171). This type of equivalent diameter is called a physical equivalent diameter. Other examples of physical equivalent diameters are listed in Table 3.5. Table 3.5. Physical equivalent diameters diameter of

formula 

sphere with same terminal velocity of free fall in a fluid

dST =

to be measured is the: 18 · · c (after Stokes)  · g

dN = 0.33 sphere with same light scatter property as real particle

where g c F S

velocity of sedimentation

F · c2 (after Newton)  · g

see Section 3.3.3

light scattering

dynamic viscosity of fluid in Pa · s gravitational acceleration in m · s−2 terminal velocity of free fall in m · s−1 density of fluid in kg · m−3 density of particle in kg · m−3  = S − F

3.1.5 Speciˇc Surface Area Specific surface area is another geometric quantity that can be used to characterize the size of individual particles, or to quantify the amount of internal surface area surrounding the hollow pore spaces within a disperse system, such as with bulk granular materials like agricultural grain in a bin, sugar crystals in a bowl, and powder in a box. The precise definition of specific surface area is slightly different for each of these two different applications. Therefore, each application will be treated separately in the following subsections. 3.1.5.1 Speciˇc Surface of Individual Particles The specific surface of an individual particle can be based on either volume or mass of the particle. Table 3.6 gives the definitions mathematically.

3.1 Particle Size

81

Table 3.6. Specific surface of particles name

formula

SI unit

specific surface based on volume

AV =

A V

m−1

specific surface based on mass

Am =

A m

m2 · kg−1

With the density of the particle known S the terms can be converted from one to the other: AV = S · Am where AV Am A V m S

(3.1)

specific surface based on volume in m−1 specific surface based on mass in m2 · kg−1 surface area of all particles of the sample in m2 volume of all particles of the sample in m3 mass of all particles of the sample in kg density of particle in kg · m−3

Whereas the specific surface based on mass is dependent on the density of the material of which the particle is made, the specific surface based on volume is a geometric quantity solely, and is independent of any information about the particle material. By sorption measurement, i.e. by measuring the ability of the particle material to adsorb a reference material at its surface, the specific surface based on mass can be obtained (see Section 1.2.5). 3.1.5.2 Speciˇc Surface Area in Bulk Materials In bulk materials made up of disperse systems, the specific surface area is defined as the total surface area covering the inner surface of all the hollow pore spaces within the porous bulk material per unit of bulk volume or unit of bulk mass of the bulk material. When the individual particles are rounded bodies of near sphere-like shape, we can assume the particles touch each other at only one point, which occupies no area. In this situation, the surface area covering the inner surface of all the hollow pore spaces can be assumed to be the same as the sum of the outer surface area of all the particles within the specified unit of volume or mass of the bulk material. The specific surface area in bulk granular materials can also be thought of as the total inner surface area exposed to a gas or fluid flowing through the porous material. In fact, specific surface is an important property that governs the degree of permeability (ease of fluid flow through pores) of a porous bulk material.For example,measurement of gas flow rate and pressure drop through

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3 Geometric Properties: Size and Shape

a powder sample enables calculation of the gas permeability of the powder, and in turn, the specific surface area per unit of bulk volume if porosity of the powder is known. This method of gas permeability is most useful to estimate the specific surface area in a sample of bulk material like agricultural grains and silage, ground meals, food powders and powdered ingredients. A more commonly used method for estimating the specific surface area within a bulk material or disperse system is to first estimate the specific surface of representative individual particles on the basis of their mass (m2 · kg−1 ), and multiply by the bulk density of the bulk material (kg · m−3 ). The result will be specific surface area of the bulk material (m2 · m−3 ). For particles with irregular shape the specific surface area can be estimated using equivalent diameters for the individual particles: A AV = (3.2) V using equivalent diameters from Table 3.4 it is (3.3) A =  · dA2  (3.4) V = · dV3 6 and so  · d2 6 · dA2 (3.5) AV =  A = dV3 · dV3 6 If precise equivalent diameters are not available, the assumption can be made that they can be approximated by assuming the particles have the shape of spheres. Then with dA = dV the calculation is simplified to: 6 AV = (3.6) d Example 3.1. Specific surface area of powdered sugar Assuming sugar powder consists of uniform spherical particles with a diameter of d = 10 ‹m with equation (3.6) we have 6 6 AV = = = 6 · 105 m−1 d 10 · 10−6 m when the particle density is S = 1500 kg · m−3 then with equation (3.1) AV 6 · 105 m−1 = = 400 m2 · kg−1 S 1.5 · 103 kg · m−3 This is the specific surface of individual sugar crystal particles in the powder. To determine the specific surface area of the powdered sugar as a bulk material (disperse system), we would multiply the solid particle specific surface by the bulk density of powdered sugar of e.g. B = 700 kg · m−3 the result would be Am =

Am =

AV 6 · 105 m−1 = ≈ 860 m2 · kg−1 S 7 · 102 kg · m−3

3.1 Particle Size

83

3.1.6 Particle Shape and Size for Crystals Some examples of varying shape and size of agricultural products have been shown in Figure 3.6. The quantities of size and shape can be quality parameters for fruits and vegetables, as well as for grains and all kind of powders. Many food powders are made up of particles that are individual crystals, such as sugars and salts. These small crystals possess various geometric forms and shapes, depending on their chemical structure and method of crystallization. When describing crystalline materials, an important distinction must be made between shape and form of a crystal. The form is a consequence of the solid state structure of the material. For example, a crystal may have an octahedral, tetrahedral or pyramidal form. Let us take as an example a crystal with a pyramidal form like in Figure 3.6, on the right: There are three geometric dimensions called a, b, c which describe the size of the pyramid. A pyramid can be a tall narrow pyramid or a short broad pyramid, based on the relative values of these geometric dimensions. A very tall pyramid looks like a needle and a very short pyramid can look like a nearly flat plate that is slightly thicker at the center than at the edges. Thus, both pyramids have the same form of a pyramid. But, they are of different shape; one is tall and the other short. This is what crystal shape means. The crystal shape can be influenced by processing. So for our example the shape of the crystal may depend on crystallization solute, temperature and ingredients. These may dictate whether our pyramid comes out nearly flat or needle like. When the solid state lattice is governed by the chemical composition of a solid material, then the form given to the crystal (pyramidal or octahedral, etc.) from the lattice is fixed and is independent of processing. On the other hand there are many solids which may occur in different solid states i.e. different lattices and therefore different forms. With good understanding and control of crystallization processes, the solid state can be controlled such that the crystal form derived from the lattice structure can also be a function of process parameters [2]. So the overall form and shape of crystalline particles is a consequence of different factors like chemical structure and conditions of crystallization. Examples where food engineers have to control crystallization are in the solidification of fat products like chocolate and butter, solidification of products rich in sugar, and freezing and hardening of aqueous solutions and suspensions involving formation of ice crystals. Nevertheless, different solid states of

Figure 3.6. Examples of particle form (cylindrical, ellipsoidal, pyramidal)

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3 Geometric Properties: Size and Shape

the same material can have different properties, such as sweetness or solubility (e.g. alpha- and beta-lactose) [16,138–140]. 3.1.6.1 Form Factor { Sphericity For food technologists who use these crystalline products as food ingredients only, there is little need to distinguish between different factors of shape and form. In that case the “overall shape” of all the particles in the bulk ingredient is the quantity needed to characterize the quality of the bulk material. This can be done with the use of descriptive terminology, such as “needle like,” “flaky,” “caking,” or with a mathematical quantity like a “form factor.” A form factor is a dimensionless ratio indicating the similarity of the given particle shape to a perfect sphere. For many applications in food engineering, this factor is called “sphericity,” meaning sphere-like. There are different definitions in use for quantifying sphericity or form factor. In general all form factors compare one property of the given particle to that same property in a perfect sphere.Among the simplest definitions are those that quantify sphericity as a dimensionless ratio of any specified equivalent diameter for the particle (arithmetic or geometric mean, or other equivalent diameter from Table 3.4 divided by the particle’s major diameter (see Table 3.3). Note that the major diameter of a particle is the same as the diameter of the smallest circumscribing sphere (smallest sphere that can completely surround the particle). Sphericity =

de dc

(3.7)

where de any specified equivalent diameter dc diameter of smallest circumscribed sphere (particle major diameter) Other more elegant definitions are given in Table 3.7. When the definition of a form factor contains the“diameter x,”this diameter can be any one of the various equivalent diameters defined earlier in Table 3.4 or Table 3.5. Therefore, it is necessary to specify which equivalent diameter has been used for calculation. Focussing on the Wadell sphericity we see its value ranges from 0 to 1. For spherical particles 'Wa = 1, for needles 'Wa → 0. Like the particle size, the particle shape is not the same for all particles but is distributed. Roundness. When a particle is observed by optical means, we see the projection of the particle (projected area). To estimate the shape of the particle from that two-dimensional view there are different roundness factors that can be used. For example, one of the most useful definitions for the roundness factor is [133]. Ap R= (3.8) Ac

3.1 Particle Size

85

Table 3.7. Definitions of form factors (examples) name

definition

formula

diameter of a volume-equivalent sphere diameter of area-equivalent particle

V,A

sphericity after Wadell

surface area of a volume-equivalent sphere surface area of particle

Wa

Heywood factor

specific surface area measured specific surface area of a sphere with diameter x

f =

=

dV dA

=

 · dV2  · dV2 = A  · dA2

AV 6/x

volume of particle measured volume form factor volume of a sphere with diameter x

kV =

V x3

surface of particle measured surface form factor surface of a sphere with diameter x

kA =

A x2

Table 3.8. Examples of sphericity [119] Particle sphere cylinder octahedron cylinder cube tetrahedron needle corn kernel sucrose crystal wheat grain oat grain soy bean potato

with R Ap Ac

sphericity after Wadell 1.000 0.874 0.846 0.832 0.806 0.670 →0 0.655 0.848 0.833 0.555 0.860 0.780

roundness factor largest projected area of particle area of smallest circumscribed circle

When particles are observed by optical means (image analysis, microscope, see Figure 3.5) to obtain a two-dimensional image (projected area), the orientation of the particles under observation will result from the natural tendency of the particle to lay in a stable position. Therefore, the roundness factor relies on a “natural rest position” that may present a different projected area from

86

3 Geometric Properties: Size and Shape

particle to particle for particles of irregular shape. In these types of situations, roundness should be measured on a large number of samples in order to determine a representative average roundness factor for all the particles in a population sample. Example 3.2. Equivalent diameter of a sphere with same volume Calculate the volume equivalent diameter of a hypothetical sphere (a) for a cube with (edge) length of a = 0.7 mm (b) for a tetrahedral regular shape with an (edge) length of a = 0.7 mm (c) for an octahedral regular shape with an (edge) length of a = 0.7 mm Solution: General: for a hypothetical sphere the volume is  V = dv3 6 so  3 6 dv = ·V  (a) cube: the volume is V = a3    3 6 3 6 3 6 3 dv = ·V = ·a = ·a     3 6 · 0.7 mm = 1.2407 · 0.7 mm = 0.87 mm dv =  √

· 2 (b) tetrahedron: the volume is V = a 12  √  √ 2 a3 · 2 3 6 3 3 6 dv = ·V = · = ·a   12 2 √ 2 3 · 0.7 mm = 0.6083 · 0.7 mm = 0.43 mm dv = 2 3

√

(c) octahedron: the volume is V = a ·3 2   √  √ 3 2 a · 2 3 6 3 2· 3 6 dv = ·V = · = ·a   3   √ 2 3 2· · 0.7 mm = 0.9656 · 0.7 mm = 0.68 mm dv =  3

3.2 Particle Size Distributions

87

Example 3.3. Equivalent diameter of sphere with same area Calculate the surface equivalent diameter of a hypothetical sphere (a) for a cube with (edge) length of a = 0.7 mm (b) for a tetrahedral regular shape with an (edge) length of a = 0.7 mm (c) for an octahedral regular shape with an (edge) length of a = 0.7 mm Solution: General: for a hypothetical sphere the surface area is A =  · dA2 so

 dA =

A 

(a) cube: the surface area is A = 6 · a2    A 6 · a2 6 dA = = = ·a     6 dA = · 0.7 mm = 1.382 · 0.7 mm = 0.97 mm  √ (b) tetrahedron: the surface area is A = 3 · a2 √ √  A 3 · a2 3 dA = = = ·a    √ 3 · 0.7 mm = 0.7425 · 0.7 mm = 0.52 mm dA =  √ (c) octahedron: the surface area is A = 2 3 · a2  √  √  A 2 3 · a2 2· 3 dA = = = ·a     √ 2· 3 · 0.7 mm = 1.050 · 0.7 mm = 0.74 mm dA = 

3.2 Particle Size Distributions Mathematical distributions play an important role in characterizing the variability of particle size that is needed in designing the processes for many biological materials that are made up of dispersed systems. Quantities of various characteristics of individuals within a population, like the age of individual

88

3 Geometric Properties: Size and Shape

persons in a population of people, or the yield of an agricultural crop in different sections of the growing field, or the amount of traffic on the internet, can be expressed by distribution functions and their related statistical parameters. One basis of a mathematical distribution given in the appendix of this book is an example of cars driving with different velocities. Highway patrol officers conducting a survey with a radar “speed gun” are taking a sample of the situation by measuring the velocity of a number of individual cars driving by the survey point. By applying statistics to the measured data, information can be given about the population of cars driving by, their velocities, percentages of different types of vehicles (cars, trucks, vans, those pulling trailers, etc.). This example leads directly to the velocity distribution of atoms or molecules in gases, which is one of the most important distribution functions for many diffusion processes found in nature and in food technology (see Appendix 15.2). Let us now focus on particle size distributions. For mathematics it is not relevant whether we have a natural product like corn cornels or a manufactured product like powdered sugar or flour. More important is the quantity which is measured to get information about the size of the particles. This quantity can be a length but also an area or a volume. Let us take the example where the length is measured. Because the individual particles can vary widely in size, the length of a large number of particles has to be measured. After these measurements are obtained, the resulting data are sorted into classes and further evaluated (see below). Normally, the results are frequently presented in the form of a diagram, called a particle size diagram. The horizontal axis is used for plotting the different lengths observed for the various particle sizes. The vertical axis is used to plot the information gathered from evaluation of the data. So the vertical axis depends on the type of set (category of characteristic) that was chosen to be measured. For example, this could be the percentage of population of particles measured that fell into different sizes (lengths). Very often particle sizing is done by sieving, i.e. classifying the size of particles by measuring the quantity that can fall through the openings in a given sieve (screen).A collection of sieves having different size openings (mesh size) would normally be used. The sieves would be stacked upon each other, with largest sieve mesh size at the top, and the smallest at the bottom (Figure 3.7). A pre-weighed sample of bulk particulate material would be placed on the top sieve, and the particles would be allowed to fall through the series of sieves while the entire stack is made to vibrate mechanically over a period of time. Afterward, when the stack of sieves is disassembled, the quantity of particles left on each sieve would consist of only those particles small enough to fall through the openings from the sieve above, but too large to fall through the openings of the sieve on which they rest. After classifying the size of particles in this way, the quantity of particles on each sieve is weighed to get information about the size distribution of all the particles. In this example, the weight of sample in each size category was the quantity measured in order to get a distribution about particle size, and the quantity under investigation was the size (length). The resulting distribution

3.2 Particle Size Distributions

89 Figure 3.7. Sieving machine. 1: engine housing, 2: vibrating plate, 3: set of sieves (sieve tower), 4: straps for mounting of sieve tower, 5: pan, 6: sieve 7: cover

curve will have the particle size on the x-axis and a term related to the mass on the y-axis. When the particles are not weighed, but instead are counted, then the scale on the y-axis would be a term related to the number of particles instead of mass. In order to avoid confusion with different types of size distributions, care should taken in addressing questions like “what is the quantity we want to know about (size)?” and “what is the quantity we intend to measure (weight or number)?” Table 3.9 shows a list of sets (categories of characteristics) which can be used for experimental investigation of particle size distributions. To understand this chapter fully we start with a simple case of sieving, in which the mass is measured to obtain a size distribution. Table 3.9. List of sets (categories of characteristics) used in quantifying particle size distributions sets (categories of characteristics) Index r usage number r=0 very often length r=1 very seldom area r=2 often volume r=3 often mass r = 3∗ very often ∗ Mass and volume have the same index because from a mathematical point of view they are different only by a factor called density (constant of proportionality)

3.2.1 Sizing by Sieving A powder sample has passed through a couple of sieves with different mesh sizes. Part of the sample is remaining on each sieve because the openings were too small for the particles to pass through. Now each sieve is weighed.

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3 Geometric Properties: Size and Shape

After subtracting the empty weight of each respective sieve, the fraction of the sample having particles too large to pass through the sieve is obtained. The particles of that fraction have a size larger than the mesh size of the sieve they were lying on (lower interval limit) and a smaller size than the mesh size of the upper sieve through which they fell (upper interval limit). The interval is xi = xi − xi−1 (3.9) xi + xi−1 xi = (3.10) 2 When a powder sample passes a set of different sieves, fractions of the sample will remain on some, if not each, of the sieves.A fraction is characterized by the sieve mesh size of the upper sieve and the lower sieve. Particles in the fraction are smaller than the mesh width of the upper sieve and larger than the mesh size of the lower sieve. So the size of the particles in that fraction is between the interval limits xi−1 and xi . The fraction can be further characterized by the arithmetic mean of the interval xi and by the interval width xi (see Figure 3.8, and Table 3.10 for the terms). Since the fraction, itself, is a mass fraction, then the set or category of characteristic we are using is mass, or a mass set.

Figure 3.8. Particle sizing: Definition of a category i, arithmetic mean xi and an interval width xi Table 3.10. Terms to define a set, or characteristic category xi−1 xi xi xi i

lower limit interval (mesh width of underlying sieve) upper limit interval (mesh width of upper sieve) interval width arithmetic mean of interval identity of any given interval (usually defined by upper limit)

Now the particle size distribution diagram can be drawn. The horizontal axis is the axis of particle size (a length axis). On the vertical axis is, in case of sieve analysis, the relative mass fraction of the powder sample: Q3 (xi ) =

mass fraction (xmin · · · xi ) total mass (xmin · · · xmax )

(3.11)

3.2 Particle Size Distributions

91

According to Table 3.9, in case of sieve analysis, the index of Qr is 3. On the vertical axis, Q3 can be replaced by its derivative with respect to length, x, called q3 . It is: qr =

dQr dx

(3.12)

so with r=3 q3 =

(3.13) dQ3 dx

(3.14)

in the case of sieves with intervals xi q3 (xi ) =

Q3(xi ) xi

(3.15)

with Q3(xi ) = Q3 (xi ) − Q3 (xi−1)

(3.16)

and xi = xi − xi−1

(3.9)

Example 3.4. Evaluation of a sieve analysis A powder sample was placed on the top of a set of eight sieves, and sieved for 2 minutes. By weighing the quantity of sample on each sieve, the distribution of mass fractions on the sieves could be obtained. From these data of mass fractions, the diagram showing particle size distribution of the sample can be constructed. Following the numbers in Table 3.11, starting with sieve 1, i.e. i = 1: Table 3.11. Evaluation of a sieve analysis (example). The illustration on the far left column represents each of the respective sieves (size category) i

8 7 6 5 4 3 2 1

mesh width xi /‹m 500 400 315 200 125 100 63 50 0

xi /‹m

mi /g

100 85 115 75 25 37 13 50

0.6 2.2 12.2 34.8 17.0 25.2 4.9 3.1

mi m 0.006 0.022 0.122 0.348 0.170 0.252 0.049 0.031

Q3,i

q3,i /‹m−1

1.000 0.994 0.972 0.850 0.502 0.333 0.080 0.031 0.000

0.6 · 10−4 2.6 · 10−4 10.6 · 10−4 46.4 · 10−4 68.0 · 10−4 68.1 · 10−4 37.7 · 10−4 6.2 · 10−4

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3 Geometric Properties: Size and Shape

mi mi m

mass fraction in category i in kg relative mass fraction in category i

mi =

i 

mi

sum of mass fractions with size x < xi “sum particles passing through”

mi

total mass of sample in kg

i=1

m=

k  i=1

Q3 (xi ) =

i 

mi i=1 m

mi q3,i = m xi indices i k

“ratio of sum of mass fractions passing through over total mass”“relative pass through fraction” (dimensionless) “relative pass through fraction” related to interval width (discretized derivative function of Q3 ) index of category, a sieve category is specified by the width of the upper interval limit total number of categories

Figure 3.9 shows the size distribution diagram of Q3 and q3 versus particle size x. When a set of eight sieves is used, the resulting diagram consists of eight vertical bars. We call that type diagram a histogram. A larger number of categories would result in a larger number of bars. An infinite number of categories would result in a smooth curve, as indicated in Figure 3.9. Most laboratory instruments designed for measurement of particle size distributions come with computer software, which calculates smooth curves for better presentation and comparison. The food engineer should always be aware of how many data sets upon which the reported result is based. As shown, a particle size distribution function can be calculated from a set of mass fractions obtained by weighing the sample fraction captured on each sieve. There are a few other ways for obtaining a particle size distribution like that. Recall from Table 3.9, instead of a mass fraction, a fraction of volume, surface area, length or number can be measured to obtain a distribution function. Very often the number is the quantity of measurement. The relative contribution (“relative amount”) in a general form is part of the set (xmin · · · xi ) (3.17) total set (xmin · · · xmax ) and its derivative part of set (xi−1 · · · xi ) (3.18) qr (xi ) = total set (xmin · · · xmax ) · interval width or relative part of set (xi−1 · · · xi ) qr (xi ) = (3.19) xi Then, the particle size distribution curve is constructed by Qr (x) or qr (x) versus size x. The shape of these curves is the same as that shown in Figure 3.9. Qr (xi ) =

3.2 Particle Size Distributions

93

Figure 3.9. Particle size distribution curves. In case of a sieve analysis, Q3 (x) or q3 (x) is plotted versus size x

By definition, Qr (x)ranges from 0 to 1. Mathematically, the integration of the function qr (x) for all particle sizes x yields unity (1) or 100%: xmax

qr (x) · dx = 1

Qr (xmax ) =

(3.20)

xmin

Consider now, a particle size distribution which is obtained by a counting method. Imagine we separate a bulk material into different categories of size. Then we count the particles in each category instead of weighing. In that case, r = 0 and equations (3.17)–(3.19) are valid for obtaining Q0 (x) and q0 (x), and the distribution diagrams might have a shape like that shown in Figure 3.9.

94

3 Geometric Properties: Size and Shape

Example 3.5. Comparison of simple size distributions obtained by weighing and counting Table 3.12. A 100 g powder sample was sieved, and the fractions are weighed i

xi /mm

1 2 3

0 2 4 6

xi /mm

2 2 2

xi /mm

1 3 5

mi /g

40 50 10

mi m 0.40 0.50 0.10

Q3,i

q3,i /mm−1

0 0.40 0.90 1.00

0.20 0.25 0.05

Results show that 90% of the mass of the particles has a size below 3 mm. Table 3.13. The same sample sieved and the particles in the fractions are counted i

xi /mm

xi /mm

xi /mm

Ni

Ni N

1 2 3

0 2 4 6

2 2 2

1 3 5

300 550 150

0.30 0.55 0.15

Q0,i

q0,i /mm−1

0 0.30 0.85 1.00

0.15 0.28 0.075

Results show that 85% of the (number of) particles have a size below 3 mm. Question: How much of the sample has a particle size below 3 mm? Answer: 90% of the powder mass, and 85% of all particles (in number). Example 3.5 shows that a different result will occur depending on whether the particle size distribution was measured by weighing or by counting. Fortunately, there are mathematical tools for easily transforming, e.g. a Q0 (x) function into a Q3(x) function. When Q0 (x)is plotted versus any quantity, the distribution often is called a probability distribution. Particle sizing instrumentation often is based on weighing (r = 3) or counting. Light scattering experiments are based on volume, such as in the measurement of sedimentation in a fluid (r = 3) (experimental; see Section 3.3). The principles learned in this section are valid not only for size distribution. With these same principles the distributions of many other characteristics of a population of individual entities can be obtained, such as form factor distributions, velocity distributions, age distributions. Not to be confused with the different types of sets (categories of characteristics) that can be measured, different types of diagrams, and the large number of quantities which must underlie the obtaining of valid distributions,here are some rules to be followed:

3.2 Particle Size Distributions

95

Table 3.14. Rules for preparing a distribution function do put on the x-axis find out r

this the quantity which is distributed what was measured?

put on the y-axis

Qr or qr

name the distribution after the quantity on the x-axis

example size answer: weight of fractions so (r = 3) Q3 size distribution

Quantiˇcation of Distribution Functions The simplest way of reading a distribution diagram and to communicate information about a sample distribution is to use statistical quantities that can be used to quantify the distribution of a characteristic in a population, like median, modal, average particle size, specific surface area, or equivalent diameter based on specific surface area (Sauter diameter). Each of these is described in greater detail in the following subsections: 3.2.2 Median The median value of a distributed characteristic in a population set is the middle value of the range of values for that characteristic within that set. The median can be determined by listing or sequencing the values of the characteristic into an increasing order (from smallest to largest). Then, the value in the middle of the sequence is the median. So the median cuts the population into two equal parts. Because of this, the median is often given the mathematical symbol x50. If the number of values n in the sequence is an even number, then two values will be at the middle of the sequence. In that case, the median is the average of these two values. Table 3.15 shows the definition for the median in mathematical terms: To calculate a median, there is no need for knowledge about the type of distribution. Compared to e.g. the arithmetic mean, another advantage of the median is that it is robust against singular extreme values. Example 3.8 demonstrates this. Table 3.15. Definition of median n

definition of median

n = odd number

x50,r = x n+1 2

n = even number

  x50,r = x n2 + x n2 +1 /2

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3 Geometric Properties: Size and Shape

Example 3.6. Median in a sequence having an odd number of values 48

51

55

60

66

75

87

The median is 60. Example 3.7. Median in a sequence having an even number of values 48

51

55

60

66

75

76

87

The median is 63 (average between 60 and 66). Example 3.8. Median in a sequence of values having an extreme at one end 48

51

55

60

66

75

76

187

The median is 63, which is more representative of the population than if an arithmetic average were to be used because of the extreme singular value at the end of the sequence. From a diagram like in Figure 3.9, the median can be read by finding the 50% percentile, and reading the related value on the x-axis. A median has to carry information from what type of set it was taken (r = 1,2 or 3). The median of a sieve analysis (r = 3) would be written as: x50,3 Example 3.9. Interpretation of a median The median of a sieve analysis is x50,3 = 75 ‹m.That means 50% have a particle size smaller than 75 ‹m. So, 50% of the mass of the sample will pass a sieve with a mesh width of 75 ‹m. It does not mean that 50% of all the particles (of the number of particles) are smaller than 75 ‹m. 3.2.3 Modal Value The modal value xh,r characterizes the maximum of the qr (x) function. It can be thought of as the maximum in a frequency distribution function. If the distribution is a q0 (x) function, then the maximum indicates the particle size which occurred most frequently (in the case of number of particles counted,

Figure 3.10. Examples of distribution curves: mono-modal (left), bi-modal (right)

3.2 Particle Size Distributions

97

with r = 0). In the case of the q3 (x) function, the maximum indicates the particle size which contributes most to the mass of the sample (in the case where the particle samples were weighed, with r = 3). The modal value can be obtained mathematically by calculating the derivative of the Q(x) function, and then finding the point of zero slope. If the distribution is presented as a histogram in the form of a bar graph, then the highest bar indicates the modal value. Distribution curves Q(x) sometimes have more than one maximum (see Figure 3.10). In that case it means there are two or more particle sizes which have the most significant contribution to the mass of the sample set measured. In Figure 3.11 the difference between the modal value and median is illustrated. 3.2.4 Average Particle Size { Integral Mean An alternative characteristic value that can be used to quantify a distribution function is the average, or arithmetic mean. To obtain the average diameter in a population of particles with different diameters,all the individual particle sizes that were measured are added and the resulting sum divided by the number of added values. Recall that the median and modal value do not carry any information about each of the values measured. In contrast, it is important to note that the average value depends heavily on the value of each of the individual measurements, and when a measured value occurs n times, it will contribute to the average n times. So an average can be calculated by multiplying each value with a factor which accounts for the frequency of contribution of it. This is often called a “weighing factor” that is assigned to each of the individual contributions. In this case, the average value is sometimes called a “weighted average.” To understand the mathematical approach for obtaining such a weighted average, we will refer back to our previous sieve analysis again: For a given average particle size (arithmetic mean of particles in a given size category): xi + xi−1 xi = (3.10) 2 we have a fraction, i.e. a mass fraction mi (3.21) = Q3,i m The contribution of xi to the average is xi · Q3,i

(3.22)

so for all particle categories, x3 =

k  i=1

Because of

xi · Q3,i

(3.23)

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3 Geometric Properties: Size and Shape

Q3 = q3 · x

(3.24)

this is x3 =

k 

xi · q3,i · xi

(3.25)

i=1

This result is the weighted average particle size derived from a q3(x) function.It is the sum of all size values measured xi multiplied with their related“weighing factors” (relative contribution) q3 · x. As we further diminish the size range of each category from the extended sieve categories x to infinitely small intervals dx, we can express the summation as an integral: xmax

xi · q3,i · dx

x3 =

(3.26)

xmin

This is the true weighted average of the particle size distribution of our sample. The index 3 in x3 indicates that we are still in the example of our sieve analysis (r = 3). In a general form the weighted average (integral mean) is: with a general fraction

i = Qr,i

(3.27)

the contribution of xi to the average is xi · Qr,i

(3.28)

So, for all particle classes it is xr =

k 

xi · Qr,i

(3.29)

i=1

because of Qr = qr · x

(3.30)

that is k 

xi · qr,i · xi

(3.31)

with x → 0 xmax xr = xi · qr,i · dx

(3.32)

xr =

i=1

xmin

or more generally

3.2 Particle Size Distributions

99

xmax

x · qr · dx

xr =

(3.33)

xmin

This is the general definition of an average calculated from a distribution function qr (x). Referring to Figure 3.9,we can understand the step x → dx as progressing to more and more ever smaller intervals. This process can be envisioned by beginning with a bar-graph histogram with only three or four relatively wide bars. Then, imagine that these few wide bars are further subdivided into a greater number of narrower bars to show the distributions over a greater number of size categories with each of narrower range. Eventually, we can produce a histogram with thousands of bars, and ultimately we see the smooth curve of q(x) instead of discrete steps. Some particle sizing software provides automatic calculation and conversion of different characteristic values used to quantify particle size distributions (mean, median, modal, etc.). So, let us look at what can be done starting with an average particle size (arithmetic mean). Let us start with a q0 (x) distribution. So the type of set (or population) is “number of particles:” Equation (3.32) now is: xmax

x · q0 (x) · dx

x0 =

(3.34)

xmin

That is the “normal” arithmetic mean of the particle size, meaning the particle length which was measured. Often it is written with the notation x = x0 . Suppose for a given task, we do not need the arithmetic mean of the length, but the mean of the length squared (representing area) or cubed (representing volume), instead. Then, we can use: xmax

x02

=

x2 · q0 (x) · dx

(3.35)

x3 · q0 (x) · dx

(3.36)

x4 · q0 (x) · dx

(3.37)

xmin

and xmax

x03

= xmin

and also xmax

x04 = xmin

How do we interpret such average quantities?

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3 Geometric Properties: Size and Shape

Let us take (3.36). The equation provides the average of all x cubed. All x cubed are summed after multiplying each with its appropriate weighing factor to reflect the relative importance of contribution, called q(x). Assume this information about relevance was achieved by a particle counting technique (r = 0). So the contribution here is related to the number of occurrences. Recall the following: xmax Q0 (xmax ) = q0 (x) · dx = 1

(3.38)

xmin

The same rules can be applied for a q1 distribution (r = 1): Then, we have: xmax x1 = x · q1 (x) · dx

(3.39)

xmin xmax

x12

=

x2 · q1 (x) · dx

(3.40)

x3 · q1 (x) · dx

(3.41)

xmin xmax

x13

= xmin

and for r = 2: xmax x2 = x · q2 (x) · dx

(3.42)

xmin xmax

x22

=

x2 · q2 (x) · dx

(3.43)

x3 · q2 (x) · dx

(3.44)

xmin xmax

x23

= xmin

and for r = 3: xmax x3 = x · q3 (x) · dx

(3.45)

xmin xmax

x32

x2 · q3 (x) · dx

= xmin

(3.46)

3.2 Particle Size Distributions

101

xmax

x3 · q3 (x) · dx

x33 =

(3.47)

xmin

These types of averages are called weighted averages, and they are mathematically named the integral mean. In general, an integral mean can be defined mathematically as: xmax

xk · qr (x) · dx

xrk =

(3.48)

xmin

with k r

power of particle size x index of set used (see Table 3.9)

In statistics, these averages are called statistical moments M. Mk,r = xrk

(3.49)

In using this nomenclature, the averages from equations (3.38)–(3.47) are: for r = 0 xmax

x · q0 (x) · dx

M1,0 = x0 =

(3.50)

xmin xmax

M2,0 =

x02

x2 · q0 (x) · dx

(3.51)

x3 · q0 (x) · dx

(3.52)

x4 · q0 (x) · dx

(3.53)

x · q1 (x) · dx

(3.54)

= xmin xmax

M3,0 =

x03

= xmin xmax

M4,0 =

x04

= xmin

for r = 1 xmax

M1,1 = x1 = xmin

xmax

M2,1 =

x12

x2 · q1 (x) · dx

= xmin

(3.55)

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3 Geometric Properties: Size and Shape

Table 3.16. Nomenclature of characteristic particle sizes k

r

dk,r

1

0

d1,0

2

0

d2,0

3

0

d3,0

2

1

d2,1

3

1

d3,1

3

2

d3,2

4

3

d4,3

with statistical moments M1,0 M0,0 M2,0 M0,0 M3,0 M0,0 M2,0 M1,0 M3,0 M1,0 M3,0 M2,0 M4,0 M3,0 xmax

M3,1 =

x13

=

x3 · q1 (x) · dx

(3.56)

x · q2 (x) · dx

(3.57)

x2 · q2 (x) · dx

(3.58)

x3 · q2 (x) · dx

(3.59)

x · q3 (x) · dx

(3.60)

x2 · q3 (x) · dx

(3.61)

xmin

for r = 2 xmax

M1,2 = x2 = xmin xmax

M2,2 =

x22

= xmin xmax

M3,2 =

x23

= xmin

for r = 3 xmax

M1,3 = x3 = xmin xmax

M2,3 =

x32

= xmin

3.2 Particle Size Distributions

103 Figure 3.11. Median value x50,r , modal value xmod and average xr (integral mean) of the qr (x) curve [1]

xmax

M3,3 =

x33

x3 · q3 (x) · dx

=

(3.62)

xmin

Statistical moments can be mathematically converted from one into another. For example in the case of the q0 function, the conversion is: xmax

xk · qr (x) · dx = xrk =

Mk,r = xmin

Mk+r,0 Mr,0

(3.63)

Table 3.16 shows the nomenclature used for characteristic particle sizes, and how they are formed with the help of statistical moments. Figure 3.11 shows the difference graphically between average (integral mean value) xr , modal value xmod and median value x50 in a population distribution. 3.2.5 Speciˇc Surface Distribution Recall, the ratio of surface area of a particle to its volume is called specific surface. It is: A 6 AV = = ·f (3.64) V x where specific surface in m−1 AV f Heywood factor A particle surface in m2 V particle volume in m3 x particle size in m So, for a sample consisting of these particles, we have:

104

3 Geometric Properties: Size and Shape k 

AV =

Ai

Atotal i=1 = Vtotal Vtotal

k A  i · Vi k  V Ai Vi i=1 i = = · Vtotal Vi Vtotal

(3.65)

i=1

which means AV =

k  Ai i=1

Vi

· Q3,i =

k  6·f i=1

xi

· Q3,i = 6 · f ·

With x → 0, this is xmax 1 AV = 6 · f · · q3 (x) · dx x

k  Q3,i i=1

xi

(3.66)

(3.67)

xmin

With equation (3.66) the calculation of the specific surface area is possible directly from a sieve analysis. When a q3 function is available, then equation (3.67) is the one to choose. In case of r = 0 it is k 

AV = 6 · f

x2 x3

= 6·f ·

i=1 k 

xi 2 · Q0,i (3.68) 3

xi · Q0,i

i=1

With x → 0 this is xmax x2 · q0 (x) · dx AV = 6 · f ·

xmin xmax

(3.69) x3 · q0 (x) · dx

xmin

Using statistic moments then only AV = 6 · f ·

M2,0 M3,0

(3.70)

3.2.6 Sauter Diameter The so-called Sauter diameter d32 is another equivalent diameter. It is the diameter of a hypothetical sphere with the same specific surface as the irregular shaped particle. So it is: d3,2 =

1 M3,0 · f M2,0

(3.71)

3.2 Particle Size Distributions

105

or d3,2 =

x3 x2

·

1 6 = f AV

(3.72)

Example 3.10. Calculation and comparison of characteristic particle sizes Taking the sieve analysis from Table 3.11: median modal value average particle size

x50,3 = 113 ‹m xh,3 = 110 ‹m 8  x3 = xi · Q3,i = 145 ‹m i=1

specific surface (with f = 1.0)

AV = 6 · f ·

8  Q3,i i=1

Sauter diameter

d3,2

xi

= 56432.8 m−1

6 = = 106 ‹m AV

3.2.7 Characteristics of Distributions For many process engineering applications, it is important to know “what is the mean” and “how broad” is the distribution. Options used to statistically quantify these characteristics about distributions are statistical parameters, such as the“norm” of a distribution and the standard deviation from the norm. For every day work in engineering it is useful to have linear functions instead of nonlinear curves that require more complicated mathematical equations. Particle size distributions can be transformed into linear functions by using appropriate transformation model functions. In the following subsections we will learn about some options to represent complete distribution functions by two parameters only. GGS Distribution The function after Gates, Gaudin and Schuhmann is a power-law function. By plotting values on logarithmic paper, the distribution function appears as a straight line which can be easily quantified by a slope and an intercept: for x ≤ xmax it is m  x Qr (x) = xmax so lg Qr (x) = m · lg x − m · lg xmax

(3.73)

(3.74)

When a particle size distribution function can be fitted with a GGS distribution then two parameters are sufficient to represent the total curve. These are the

106

3 Geometric Properties: Size and Shape

slope m, representing the width of the original distribution and xmax which locates the point of the curve where Qr (x) = 1. Log Normal Distribution qr (x) =

·

1 √



1 · exp − 2 2



x−x 

2  (3.75)

with the median x50,r taken for x,  as standard deviation of x and c=

x−x 

(3.76)

it is c2 1 · e− 2 qr (c) = √ 2

(3.77)

Using special grid paper a log-normal distribution appears as a straight line in the diagram. Also a particle size distribution can be plotted as a straight line which is characterized by its median x50,r and the standard deviation  of the distribution. RRSB Distribution The RRSB distribution (after Rosin, Rammler, Sperling and Benett) is a two-parameter exponential function  x n 1 − Qr (x) = exp −  (3.78) x with Rr (x) = 1 − Qr (x) Then



1 lg lg Rr



= n · lg x − n · lg x + lg(lg e) = n · lg x + d

(3.79)

(3.80)

where d is a constant. Using special grid paper (RRSB net) our particle size distribution appears as a straight line in the diagram which can be characterized by x and the slope n only. There is no guarantee that an experimental particle size distribution can be represented with one of the models mentioned. Whether a model can be used or choosing what model will best fit the data depends on the food sample properties that result from the process of its preparation, e.g. milling process or crystallization process. For more details about working with distribution functions see [1] or [17].

3.3 Measuring Particle Size by Other Techniques

107

3.3 Measuring Particle Size by Other Techniques Particle size measurement can be performed with optical, electrical, weighing or other techniques.In the following sections the principles of these techniques are described. Before a particle size measurement can be made, a representative sample needs to be taken. Because powders and bulk materials can show separation effects or particle breakage into nonrepresentative fragments (particle attrition), it is necessary to carefully arrange the conditions of sampling. Sometimes it is useful to take samples at different locations within the bulk material, or at different times from the same location, and to mix the sample prior to analysis. On the other hand there are devices for automatic dividing of a large sample into a certain number of smaller samples without changing the particle size distribution. For details, see [1]. 3.3.1 Weighing Technique The use of gravity to measure the amount of particles is the most commonly used technique for powders and bulk materials. Although the technique is widely used in laboratories, for on-line sensing during the manufacturing process on the factory floor, other techniques have to be chosen. We have already learned how sieving can be used to divide the sample into mass fractions of different size ranges. The particle amount in each fraction is measured by weighing. So particle sizing by sieving is not a measurement of particle dimensions but a gravimetric measurement of the fractions obtained elsewhere. Normally, particles are divided into size categories by sieving, but other techniques, such as light scattering, and air flow separation (air classification) can be used as an alternative to sieves for the same purpose. In case of using sieves, the mesh width is taken to describe the particle size. The arithmetic mean between the mesh widths of two sieves is taken for the average particle size in that size category (class). xi + xi−1 xi = (3.10) 2 In case of nonisometric particles, the mesh width can be a rough estimate of particle size only. Particles with different diameters in different directions will pass a given sieve to a certain extent depending on sieving conditions that can induce particle orientation. Furthermore there are different types of sieves available. For example, in the USA, there are sieves designed to comply with US standards (e.g.Tyler Sieves),and in the EU there are sieves designed to comply with EU or other standards. There are woven screen sieves or perforated plate sieves. Perforated plate sieves can have orifices with square, rectangular, round or other shapes. Because of this feature, the type of sieves used for particle size measurement should be noted and reported with the results from a sieve analysis.

108

3 Geometric Properties: Size and Shape

Performing a sieve analysis with a device as shown in Figure 3.7 consists of putting a defined (pre-weighed) amount of sample on the top of the set of sieves. By starting the sieve machine to induce mechanical vibration to the sieves, the sieves will vibrate at set frequency and amplitude of vibration for a fixed period of time. The vibrating movement often is a vertical reciprocation (up and down vibration) with adjustable amplitude and frequency. It can also be a horizontal movement (by eccentric rotation), or any combination of both types. The movement of the sieves provides each particle multiple chances for passing through an orifice. This vibrating motion also has a mixing effect and helps to avoid clogging of the sieves. To enhance these effects, in additional to the sieve movement, sieving tools like ceramic balls or cubes of rubber can be added to the powder sample on each sieve. In any case, disintegration of the particles (particle attrition) has to be avoided. For difficult sample materials that have a tendency for clogging, there are sieve machines available that make use of air impingement as a sieving tool. After operation of the sieving machine, the sieve tower is disassembled and the sieves are weighed to get the weights of the fractions. For evaluation of such weights, see Table 3.11 and Figure 3.9. For very small particles, there are sieves available for use with suspending solvents. As an additional sieving tool, ultrasonic vibration can be used. 3.3.2 Sedimentation and Aerodynamic Classiˇcation with Fluids Sedimentation in liquids and aerodynamic classification with gases (usually air) can also be used to separate particles into classes or categories. Both of these techniques are based on Stokes’ law involving the terminal velocity of a particle in free fall through the fluid. When a particle is allowed to fall freely through a fluid under the acceleration of gravity,the increasing relative velocity of the particle through the fluid will increase drag force (caused by friction at the particle–fluid surface) at an exponential rate (quadratic). Stokes’ law (equation (3.81)) states that terminal velocity is reached during free fall when the drag force on a particle acting upward becomes equal to the gravitational force on the particle (particle weight) acting downward. At that moment, the particle comes into static equilibrium, and velocity ceases to increase, having reached “terminal” velocity. The force of friction after Stokes (drag force) is FR = 6 · r · · v

(3.81)

The gravitational force (weight of body) is FG = (K − F ) · VK · g

(3.82)

At equilibrium we have steady state velocity (terminal velocity) v=

(K − F ) · VK · g 6 · r ·

with the volume of a sphere

(3.83)

3.3 Measuring Particle Size by Other Techniques

109

Figure 3.12. Sedimentation of spherical particles in a fluid

V=

4  · r3 3

(3.84)

2 · g · r 2 · (K − F ) 9·

(3.85)

Then v= or with d = 2·r v= where FG FR r d V g v K F

g · d 2 · (K − F ) 18 ·

(3.86) (3.87)

weight force in N drag force in N radius of spherical particle in m diameter of spherical particle in m volume of spherical particle in m3 gravitational acceleration in m · s−2 particle velocity in m · s−1 density of particle in kg · m−3 density of fluid in kg · m−3 dynamic viscosity of fluid in Pa · s

From equation (3.87) it can be seen that from the steady state velocity v of the particles we get information about their diameter d. This is the principle of separation by sedimentation and aerodynamic particle classification. Sedimentation Sedimentation is the process of allowing particles initially suspended near the surface of a liquid to fall freely to the bottom, forming a layer of sediment (sedimentation). Note that sedimentation relies on the influence of gravity, and is only possible when the particles have solid density greater than the suspending liquid. It can be seen from Stokes’ law that the velocity of free fall (terminal velocity) will largely depend upon particle diameter (equation (3.87)). Therefore, when a sample of particles having different sizes (different diameters) is placed on the surface of a liquid at the top of a tall container, the particles will fall to the bottom at different velocities, depending on their diameters.

110

3 Geometric Properties: Size and Shape

The largest size particles will reach the bottom first, followed by those slightly smaller, and so on. For sedimentation separation in liquids the sample of particles is placed into an appropriate liquid (having less density than the particle solid density, and having no sorption or chemical reactions at the particle/liquid surface) contained in a relatively tall vessel (to allow sufficient time for free fall). After the specified time for free fall, a sample of the sediment layer is analyzed, e.g. by optical or gravimetric measurement. In this case the particle size measured is the equivalent diameter for sedimentation (see Table 3.5). Air Classiˇcation Separation by air classification is commonly used in the process industries to separate particles having different terminal velocities (hence, particle size diameters) in air. The scientific principle is the same as sedimentation (based on Stokes’ law), but works backwards from sedimentation. Instead of allowing particles to free fall in still air, the mixture of particles is placed to rest upon a perforated screen surface. Then, air is blown upward from beneath the perforated screen at a controlled velocity.As soon as the velocity exceeds the terminal velocity of the smallest size particles, the drag force acting upward will exceed the weight of the particles, and they will become airborne, rising upward to be captured by an appropriate particle collection system. Samples of these particles can then be taken for analysis to measure particle size. When higher air flow velocities are used, then larger size particles will become airborne, and so on. In the oil seed crushing and grain processing industries, air classification is widely used to automatically separate the light weight (low density) hulls, husks and shells from the relatively higher density seeds, grains and kernels. In this way, the seeds continue down-stream to the oil extraction or other process operations free of contamination from the unwanted shells or hulls. 3.3.3 Optical Techniques Microscopic techniques, with manual or automatic image analysis and light scattering are methods for optical measurement of particle size. Light scattering often is performed with monochromatic laser light. The particles are passed along the light beam, which causes scattering by reflection, refraction and diffraction as illustrated in Figure 3.13. Following the theory of Mie, the angle of diffraction is related to the size of a particle. So, the particle size can be determined by measuring the angle of diffraction. The Mie theory assumes the particles have a spherical shape, and does not take into account for particle with other shapes. This is the reason why light-scattering experiments with nonspherical particles will give only a light-scattering equivalent diameter, i.e. the diameter of a spherical particle with the same light scattering property. Most light scattering instruments provide particle size distributions

3.3 Measuring Particle Size by Other Techniques

111 Figure 3.13. Scattering of light (1) at a particle (2) by diffraction (3), refraction (4) and reflection (5)

Figure 3.14. Principle oflaser scattering: I laser,II sample particles,III scattered light,IV lens, V focus surface, VI light detector, VII distribution of light intensity I (schematic)

based on the particle volume (i.e. Q3 (x) function), and computer software for use in making transformations to other types of distributions, e.g. Q0(x). An important advantage of light scattering is the speed of the method. Because the light intensity distribution can be measured multiple times during one second,the results,including related statistical parameters,can be obtained within seconds. Furthermore, the sample particles can be measured while they are in motion. There is no need to have the sample particles in a fixed position. So,flow cells can be mounted in the path of the light beam with sample particles flowing through it. The particles can be suspended in appropriate liquids or in air. Another advantage of light scattering is the very small particle size (10−7 to 10−3 m), which can be measured. Limits of the technique are the wavelength used and the concentration of particles. If the concentration is too high, multiple scattering occurs. For measurement of highly concentrated emulsions by acoustics see Section 12.3 and [1]. 3.3.4 Electrical Techniques Electrical techniques are also available for particle size determination. In these techniques, the particles are suspended in a liquid which contains dissolved electrolytes, and pumped through a capillary tube (see Figure 3.15). Each time a particle passes through the capillary tube, the electric impedance (see Section 8.2) of the capillary changes.This change in impedance is related to the volume occupied by the particle (volumetric size) in the capillary. These changes

112

3 Geometric Properties: Size and Shape Figure 3.15. Electrical counting of particles with different sizes through a capillary tube (schematic)

in impedance are classified by their magnitude, and automatically counted. Instruments based on this principle provide Q0 (x)distributions. To cover a wide range of particle sizes capillaries with different diameters are used. 3.3.5 Other Techniques Another possibility for estimating particle size is to measure the specific surface of a powder by e.g. gas adsorption. Once a monolayer is formed (see Section 1.2) using the BET theory the surface area of the solid can be calculated. For powders, gaseous nitrogen at 77 K is normally used. A further possibility is measuring the flow resistance of a powder sample against a gas (or air) after Blaine [4]. Acoustic techniques are based on measuring the acoustic impedance of a sample. Table 3.17. Comparison of particle sizing techniques technique sieving sedimentation, and air classification light scattering gas adsorption electrical counting microscopy, image analysis acoustic impedance

normally provides a distribution with r = 3 (mass) r = 3 (mass)

applications

r r r r

powder, bulk materials emulsions power powder powder, bulk materials emulsions, foams emulsions

= 3 (volume) = 2 (area) = 0 (number) = 0 (number)

r = 3 (volume)

powder, bulk materials powder, bulk materials

User Notes A median can be calculated from different types of distributions. To account for this, the abbreviations in Table 3.18 are usual.

3.3 Measuring Particle Size by Other Techniques

113

Table 3.18. Nomenclature of different medians formula

median of a distribution based on

x50,3 x50,2 x50,1

volume (r = 3) area (r = 2) length (r = 1)

x50,0

number (r = 0)

Table 3.19. Where to apply what diameter from distribution functions [109] diameter

this integral mean is based on . . .

so, it is useful in characterization of, e.g.

d3,1 d3,2 d4,3

. . . length . . . area . . . volume

flow through pores of membranes surface adsorption of droplets creaming of emulsions due to buoyancy

Table 3.20. Common sets of sieves and their sizes [109] DIN 4188 (mm) 0.022 0.032 0.045 0.063 0.09 0.125 0.18 0.25 0.355 0.5 0.71 1 1.18 1.4 2 2.8 4 5.6

ASTM E11-70 (mesh)

400 325 230 170 120 80 60 45 35 25 18 16 14 10 7 5 3.5

ASTM BS Tyler E161-70 410:1969 (mesh) (‹m) (mesh) 22 32 38 38 400 45 45 325 63 63 250 90 90 170 125 125 115 180 80 250 60 355 42 500 32 710 24 1000 16 1180 14 1400 12 2000 9 2800 7 4000 5 5600 3.5

Table 3.21. US standard sieve numbers and their related sizes Sieve No. 4 6 8 12 16 20 30 40 50 70 100 140 200 270 pan

nominal sieve opening in mm in inch 4.76 0.187 3.36 0.132 2.38 0.0937 1.68 0.0661 1.19 0.0469 0.841 0.0331 0.595 0.0234 0.420 0.0165 0.297 0.0117 0.210 0.0083 0.149 0.0059 0.105 0.0041 0.074 0.0029 0.053 0.0021

114

3 Geometric Properties: Size and Shape

3.4 Applications chocolate: particle size and quality peanut butter: particle size and quality fibers of grain: particle size and sorption organoleptic testing: particle size and creaminess apple powder: particle size and functional properties separation of hulls from seeds in grain processing separation of cream from skim in milk processing sizing of protein nanofibers

[10,11] [12] [13] [14] [15] [117] [109] [18]

Literature 1. Stieß M, Ripperger S (2007) Mechanische Verfahrenstechnik. Springer, Heidelberg 2. Hartel RW (2001) Crystallization in foods. Aspen Publishers, Gaithersburg 3. Baltes W (Hrsg.) (1995) Schnellmethoden zur Beurteilung von Lebensmitteln und ihren Rohstoffen. Behr‘s, Hamburg 4. Allen T (1997) Particle Size Measurement. Chapman and Hall, New York 5. Coupland JN, McClements D (2001) Droplet size determination in food emulsions: comparison of ultrasonic and light scattering methods. J Food Engineering 50:117–120 6. Javanaud C, Gladwell NR, Gouldby SJ, Hibberd DJ, Thomas A, Robins MM (1991) Experimental and theoretical values of the ultrasonic properties of dispersions: effect of particle state and size distribution. Ultrasonics 29:331–337 7. McClements DJ, Povey MJW, Jury M, Betsanis E (1990) Ultrasonic characterization of a food emulsion. Ultrasonics 28:266–272 8. Povey MJW, McClements DJ (1988) Ultrasonics in food engineering, Part I: introduction and experimental methods. J Food Engineering 8:217–245 9. Babick F, Ripperger S (2002) Schallspektroskopische Bestimmung von Partikelgr¨oßenverteilungen submikroner Emulsionen. Filtrieren und Separieren 16, 6:311–313 10. Ziegler G, Mongia G, Hollender R (2001) The role of particle size distribution of suspended solids in defining the sensory properties of milk chocolate. Intern J Food Properties 4:353 11. Windhab EJ, Attaie H, Breitschuh B, Braun P (2003) The functionality of milk powder and its relationship to chocolate mass processing, in particular the effect of milk powder manufacturing and composition on the physical properties of chocolate masses. Intern J Food Science & Technology 38:325–335 12. Santos BL, Resurreccion AVA (1989) Effect of particle size on the quality of peanut pastes. J Food Quality 12:87–97 13. Strange ED, Onwulata CI (2002) Effect of particle size on the water sorption properties of cereal fibers. J Food Quality 25:63–73 14. Kilcast D, Clegg S (2002) Sensory perception of creaminess and its relationship with food structure. Food Quality and Preference 13 (7/8):609–623 15. Grover SS, Chauhan GS, Masoodi FA (2003) Effect of particle size on surface properties of apple pomace. Intern J Food Properties 6:1–7 16. Figura LO (1993) Physical modification of lactose and its thermoanalytical identification. Thermochim. Acta 222:187–194

Literature

115

17. Schubert Heinrich (2002) Handbuch der Mechanischen Verfahrenstechnik.Wiley-VCH, Weinheim 18. Zeeuw EA. van der, Sagis LMC, Koper GJM, Mann EK, Haarmans MT, Bedeaux D (1996) The suitability of scanning angle reflectometry for colloidal particle sizing. Chemical Physics 105: 1646

4 Rheological Properties

Rheology is the branch of physics in which we study the way in which materials deform or flow in response to applied forces or stresses.The material properties that govern the specific way in which these deformation or flow behaviors occur are called rheological properties.The Greek philosopher and scholar,Heraclit (550–480 BC) once said “ ˛ ´ ˛ ‘" ˜” (“everything flows”). In the context of physics,“flow” can be defined as continuous deformation over time, and it can be said that all materials can flow. Therefore the ability to flow is not only possessed by gases and liquids, but also by solids to a varying degree. Indeed, we all know examples of solids which are capable of continuous deformation over time (flow), like asphalt on a road surface after long term usage. It is also evident that temperature can have a strong influence on the ability of materials to flow. For example, the asphalt road surface will deform at a faster rate when carrying traffic during time periods of elevated temperatures. In this chapter, the rheology of both solids and liquids will be studied, but in the context of being at opposite ends of a continuum (continuous spectrum) of rheological behavior exhibited by matter in different forms. Because liquids and gases are the forms in which matter can flow most easily, the flow of liquids and gases will be covered in greater depth.After an introduction of ideal rheological behavior in solids and liquids, nonideal behavior is treated, and concepts like viscoelasticity will be discussed.At the end of the chapter, texture measurement and quantification in food will be discussed as an important practical application of rheology of solids to food technology.

4.1 Elastic Properties When a solid body at rest on an immovable surface is loaded with a weight (in this case, a force acting downward) it will respond with a deformation that can be characterized by a reduction in its initial height dimension. However, a solid body will deform in different ways depending upon how the force is acting upon it. Therefore, it is important to distinguish between the three different ways in which a solid body can be stressed (can have forces acting upon it): In the first instance, like in the example with the weight on the body, a body can be stressed under “axial loading” in an uniaxial direction (force acting normal to the surface upon which it is applied, causing a normal stress).

118

4 Rheological Properties

Figure 4.1. Types of loading with force F. I uniaxial, II bulk compression, III shear

Secondly, a body can be pressed upon from all directions, like when subjected to an underwater hydrostatic pressure or pressurized in a pressure chamber. This is called bulk compression loading, and results in a volume change in the solid body. In the third instance, a body can be loaded with a shear stress (when the force is acting parallel to the plane surface on which it is applied). Under a shear stress, the responding deformation is in the form of a bending or twisting of the body. Figure 4.1 illustrates these different types of loading which will be discussed further. 4.1.1 Uniaxial Stress As explained earlier, an uniaxial stress is caused by a force pushing or pulling the body in a direction perpendicular to the surface of the solid body upon which the force is acting. Pushing would cause a compressive stress on the body, resulting in an uniaxial “compression” deformation, in which the initial length of the sample would decrease (become shorter) in the direction of the applied force. Pulling would cause a tensile stress (tension) on the body, resulting in an uniaxial “extension” deformation, in which the initial length of the sample would increase (become longer) in the direction of the applied force. Mathematically, these opposite directions of uniaxial compression or tension would simply be assigned as positive (extension) or negative (compression) directions along the same uniaxial path. It is important to realize that,in addition to the inherent rheological properties of a material, the amount of deformation observed in a sample of material in response to a given applied force will also depend on the geometric size and shape of the sample. For example, under the same applied force, a tall slender sample will exhibit a much greater deformation than a short flat sample of the same material. Therefore observations of force and deformation will only reveal rheological behavior of the sample, but not the material out of which the sample is made. In order to have these observations reflect true rheological behavior of the material from which the sample is made, it is necessary to elim-

4.1 Elastic Properties

119 Figure 4.2. Differences in shape and size will result in different deformation in response to the same applied force, even though samples are of the same material

inate the interference caused by sample size and shape. This is accomplished by converting the applied force into stress, and the responding deformation into strain. The stress is obtained when we divide the applied force by the area of the surface over which it is applied. The strain is obtained when we divide the measured deformation (compression or extension) by the initial length of the sample in the direction of the applied force. When these transformations are made, the resulting strain that is observed in response to an applied stress will be the same for any sample of a given material, regardless of size or shape of the sample. These transformations are described mathematically below, and illustrated in Figure 4.2. The quotient of force over area is called stress, with units of force over area (N · m−2 = Pa) F (4.1) = A The relative change in length is called strain ": "= where F A  l l "

l l

(4.2)

force in N area in m2 stress in N · m−2 deformation in m initial length in m strain

Therefore, as stress increases on a sample of a given material, it will cause the responding strain to increase, as well. The relationship between stress and strain is one of the most fundamental rheological behaviors to be studied in solid materials. This relationship can be studied and analyzed mathematically when a simple graph is constructed on which the stress is plotted on the vertical axis with corresponding strain plotted on the horizontal axis, as shown in Figure 4.3.This type of graph is called a stress–strain diagram.When a material exhibits a linear relationship between increasing stress and strain, it is said to be a classic elastic material that follows Hooke’s law. According to Hooke’s law, the change in strain in response to increasing stress can be approximated by a linear function that will appear as a straight line through the origin on a

120

4 Rheological Properties Figure 4.3. Stress–strain diagram of a material with a distinct yield stress ReH

stress–strain diagram. Figure 4.3 shows such a linear relationship beginning at the origin, but only up to a certain point on the diagram. Within the range of values over which the stress–strain relationship is linear, the slope of the straight line can be mathematically expressed as the ratio of stress over strain. This ratio of stress over strain is known as the modulus of elasticity E, and must not be confused with the elasticity or degree of elasticity of a material. Elasticity can only be measured when a load–unload test is performed. In such a test, the sample is loaded to a given stress causing a corresponding strain. Then, the sample is unloaded by releasing the stress back to zero, and the degree to which the strain recovers is observed. In the case of an elastic deformation, the strain is completely reversible, and fully recovers. That means when the stress is taken away, the strain (deformation) goes back to zero. When such a test is repeated with the application of higher and higher stresses, an elastic limit is reached beyond which the strain no longer fully recovers. This means that after taking away the stress, part of the strain (deformation) will remain as a permanent “set.” This unrecovered strain is called a plastic deformation (see Section 4.3.9). Many of the most important rheological properties of solid materials can be extracted from a graph on which stress is plotted along the vertical axis, and strain is plotted along the horizontal axis. These are called stress–strain diagrams, like that shown in Figure 4.3. Examining this figure more closely, we can see that the straight line coming from the origin (0,0) indicates that Hooke’s law is valid in the linear region of the graph. The point at which the graph begins to deviate from a straight line is called the proportional limit (ReL ). Stresses beyond this point lead to nonlinear deformation (starting with ReL ). At the point ReH , the solid begins to flow or yield, but only temporarily, since the stress is able to increase once again with increasing strain up to a maximum level on the diagram.At the yield point, the strain is able to continue increasing with little or no increasing stress. If the stress were to be released at this point, the sample would keep its deformation as a permanent set (it was deformed by flowing, and is therefore a nonelastic or plastic deformation). At a certain point, the strain reaches a limit beyond which the sample can no longer sustain its original structure,and rupture or breakage occurs.That is the

4.1 Elastic Properties

121

point in Figure 4.3 where the curve ends. However, for most solids, this point occurs at the very maximum stress value reached during testing.In any case,the maximum stress value reached is called the maximum strength of the material, and the stress at which rupture occurs is called the rupture strength. Again, for most solid materials, maximum strength and rupture strength are often the same. Note the conventional use of terminology here, in that, whenever we refer to a specific stress where something happens (yield stress, maximum stress, rupture stress, etc.), we call it a “strength.” Many food and biological materials will give only a smooth curve on a stress–strain diagram that may still begin as a straight line, but will only gradually bend away from linearity such that no distinct yield point can be seen (mathematically, the stress–strain curve is monotone), as shown in Figure 4.4. In this case, it is difficult to determine at what point the material will make the transition from elastic to plastic behavior. In this type of situation, a special technique called the 0.2% strain limit can help. This technique proposes a technical limit of elasticity that is used to specify the stress at which the permanent strain reaches a value of 0.2%. Figure 4.4 shows the use of this technique as an example:

Figure 4.4. Stress–strain diagram of a material with a smooth profile showing no distinct yield point. Rp0.2 is the technical limit of elasticity for the 0.2 strain limit technique. The slope of the straight line portion gives the modulus of elasticity E for the material. The highest point on the curve (Rm ) gives the maximum strength of the material. B is the rupture point (rupture strength)

4.1.2 Young's Modulus Recall the three different ways in which a solid body can be stressed (can have forces acting upon it, see Figure 4.1): In the first instance, like in the example with the weight on the body, a body can be stressed under “axial loading” in an uniaxial direction (force acting normal to the surface upon which it is applied, causing a normal stress) Secondly, a body can be pressed upon from

122

4 Rheological Properties

all directions, like when subjected to an underwater hydrostatic pressure or pressurized in a pressure chamber. This is called bulk compression loading, and results in a volume change in the solid body. In the third instance, a body can be loaded with a shear stress (when the force is acting parallel to the plane surface on which it is applied). Under a shear stress, the responding deformation is in the form of a bending or twisting of the body. All three of these types of stress application can be studied with stress– strain diagrams like those shown in Figure 4.3 and Figure 4.4, and all three may show a linear region of elastic behavior from which the slope will give a value for modulus of elasticity. However, the value of this modulus will be different for the same material, depending upon which of the three ways the sample was stressed.When the modulus of elasticity is obtained from axial loading tests, it is called Young’s modulus E. When it is obtained from bulk compression tests, it is called the“bulk”modulus K,and when obtained from shear tests,it is called the“shear”modulus G. Each of these three moduli are fundamental rheological properties of a solid material. But, each modulus will have a different value for the same material. However, all three can be mathematically related to each other through a related rheological property called Poisson’s ratio. Young’s modulus will be discussed in this subsection, followed by discussions of the other two moduli and Poisson’s ratio in subsequent subsections. Recall that ideal elastic behavior is characterized by a straight line through the origin on a stress–strain diagram, and can be mathematically expressed by Hooke’s law:  =E·"

(4.3)

E is a material property named the modulus of elasticity. When the stress and strain result from axial loading conditions, this modulus is called Young’s modulus, and given the symbol (E). Young’s modulus gives an indication of easily a material can be stretched or contracted, and is sometimes referred to as “stiffness.” Experimentally, E is obtained from the slope of the stress–strain curve (see Figure 4.4): d = tan ˛ (4.4) d" If instead of stress and strain, the quantities of force and extension are used, Hooke’s law is expressed as: E=

F l =F· A l E·A F= · l l F = D · l with  E "

stress in N · m−2 modulus of elasticity in Pa strain

(4.5) (4.6) (4.7)

4.1 Elastic Properties

F A l l D

123

force in N area in m2 extension in m initial length of solid in m Hooke’s constant in N · m−1

In addition to the modulus of elasticity, stress–strain diagrams like those in Figure 4.3 or Figure 4.4 can be used to obtain many other rheological terms such as those listed in Table 4.1. Table 4.1. Terms for interpretation of  -" diagrams description

characteristic feature 

elastic range nonelastic range

d = const d"

maximum of curve technical limit plasticity rupture (breakage)

strain is completely reversible

d = const d" d =0 d"

strain is not completely reversible yield stress (flow starts)

 = max " = 0.2 %  shows sudden drop

"

maximum strength of material stress for 0.2 % strain permanent strain stress at rupture

strain continues without increasing stress strain at maximum stress strain goes to 0.2 % permanent strain at the point of rupture

The magnitude of some Young’s modulus values E are listed in Table 4.2. Table 4.2. Values of Young’s modulus of elasticity, examples [115] material ice stainless steel aluminum rubber dry spaghetti apple, raw gelatine (gel) banana

E/N · m−2 9.9 · 109 195 · 109 72 · 109 8 · 105 3 · 109 0.6 . . . 1.4 · 107 2 · 105 0.8 . . . 3 · 106

124

4 Rheological Properties

4.1.3 Bulk Modulus As an alternative to axial stress, a body can also be pressed upon from all directions, like when subjected to an underwater hydrostatic pressure or pressurized in a pressure chamber. This is called bulk compression loading, and results in a volume change in the solid body (see Figure 4.1). When a body is compressed from all directions, this causes a decrease of the body dimensions in x-, y- and z-directions in a Cartesian system of coordinates. This is called a bulk compression. There is also an elastic range of compression and a nonelastic range. A normal rule of convention is to let positive stress be considered pressure, which causes a reduction in dimensions from all directions, resulting in a reduction in volume of the sample. In contrast, negative stress would be called expansion, and causes an increase in dimensions in all directions, resulting in an increase in volume. This is called volume dilatation, and can happen when a sample is placed under vacuum, or is suddenly released from high pressure to a lower pressure (expansion). An elastic compression is a volume decrease in a solid caused by decreasing the distance between the molecules within the molecular structure of the material. Once the applied pressure is released back to zero, the initial molecular distances and the initial volume of the body are restored. Hooke’s law would again apply: −

dV =  · dp V

=−

1 dV V dp

(2.18)

dp dV

(4.8)

K = −V where V p K 

(2.18)

volume in m3 pressure in Pa bulk modulus in Pa compressibility in Pa−1

Figure 4.5. Bulk compression

4.1 Elastic Properties

125

The bulk modulus K characterizes how a material can withstand an elastic compression, and is sometimes referred to as “firmness” of a material. Normally gases can be compressed easily (see Section 2.3.2), liquids less so, and solids need much higher loads (like in high pressure processing) to show changes in volume. Whereas gases and normal liquids show an isotropic K modulus, solids may have different values in different spatial directions. Such a material property is called an anisotropic material. A bulk modulus for an anisotropic material can be expressed as a tensor with 9 components: ⎛ ⎞ Kxx Kxy Kxz K = ⎝ Kyx Kyy Kyz ⎠ (4.9) Kzx Kzy Kzz For isotropic solids and normal fluids the modulus is simply: K = Kij

(4.10)

and 1 (4.11) = K A pressure has the same value in all three directions. So normally, the reduction in the dimensions of a solid is the same in all three directions. If a material has an anisotropic bulk modulus, the reduction in dimensions can be different in each of the three directions. The consequence is a change in the shape, as shown in Figure 4.6. In high pressure processing of foods, phenomena like this should be considered.

Figure 4.6. Compression of an anisotropic solid. An isotropic material shows the same reduction in length in all directions (left). An anisotropic material (right) does not, and shows a change in shape.

Table 4.3. Values of bulk modulus K, examples [115] material ice stainless steel dough rubber glass

K/N · m−2 10 · 109 170 · 109 1.4 · 106 1.9 · 107 3.8 · 109

126

4 Rheological Properties

4.1.4 Shear Modulus When a force Ft is acting parallel to a surface, the resulting stress is called a shear stress. The type of loading is called shear. The resulting deformation is a shear deformation. The shear deformation is not an axial change in length, but a rotational change (torsion or twisting). For this reason, it can be expressed as an angle of deformation.

Figure 4.7. Shear stress on a body because of a tangential force Ft causing angular rotational strain

The quotient of tangential force Ft and area A is called the shear stress . The angle of deformation  is called shear deformation, and tangent of that angle is shear strain. In the elastic range of the material, the elongation x (see Figure 4.7) with respect to the sample height y is the ratio xy , which is tan  . This is the shear strain, and is a linear function of the shear stress in the elastic region: With Ft A in case of elasticity it is: =

(4.12)

 = G · tan 

(4.13)

With small deformations it can be approximated: tan  = 

(4.14)

 =G·

(4.15)

so Solids with ideal elastic behavior show a linear relationship between shear stress and shear strain. The material property G in equation (4.15) is called the shear modulus G,and gives an indication of how easily the material will bend or twist in response to an applied shear stress. This is sometimes called “rigidity” in common terminology. Thus, the common terms “stiffness,” “firmness” and “rigidity” can be related to the Young’s, bulk and shear modulus, respectively. Just as with the bulk elastic modulus, the shear modulus for an anisotropic material can be written as a tensor. For isotropic materials G is not dependent on the spatial direction.

4.1 Elastic Properties

Ft A  G 

127

tangential force in N area in m2 shear stress in N · m−2 shear modulus in N · m−2 angle of deformation in rad

Example 4.1. A rubber eraser (length 20 mm) with a cubic shape is loaded with a tangential force of 10 N as shown in Figure 4.7. With a shear modulus of the material of 0.2 · 106 Pa calculate the angle of deformation and the elongation x. Solution: 10 N F = 25000 Pa = = A 2 · 2 · 10−4 m2  =

25000 Pa  = = 0.125 G 0.2 · 106 Pa

x =  · y = 0.125 · 20 mm = 2.5 mm 4.1.5 Poisson's Ratio and Transverse Strain Let us return to the case of axial loading. When a sample of material is subjected to a normal stress pushing downward, it will show an elastic decrease in length l. At the same time, however, it must also show an increase in thickness d, that will appear as a “bulge“ along the sides of the sample. This is because the volume of the sample cannot change under simple axial loading. Thus, if the dimension in one direction (height) must decrease because of the applied stress, then dimensions in the transverse direction (thickness) must increase. The relative change in thickness d d is called “transverse strain” and is dependent on the axial strain. The ratio of transverse strain to axial strain produces a proportionality factor called Poisson’s ratio . These terms are defined mathematically below: "q =

d d

(4.16)

l l

(4.2)

"q = − · "

(4.17)

"=

Poisson’s ratio: transfer strain

=− axial strain

(4.18)

128

4 Rheological Properties

where d d l l " "q

thickness in m change in thickness in m length in m change in length in m axial strain transverse strain Poisson’s ratio

Table 4.4. Poisson’s ratio for different materials, examples [115]

material cheddar cheese potato issue rubber apple tissue apple copper steel glass cork

0.5 0.49 0.49 0.37 0.21–0.34 0.33 0.30 0.24 0

Table 4.5. Special cases for Poisson’s ratio case

K

E

G



A B

large small

small large

small large

= 0.5

=0

Table 4.6. Calculations of moduli case A

= 0.5

case B

=0

G=

E E = 2(1 + ) 3

G=

E 3 = K 2 2

K=

E =∞ 3(1 − 2 )

K=

E E = 3(1 − 2 ) 3

E =3·G

E =3·K

Poisson’s ratio provides a key link between the Young’s E, shear (G) and bulk (K) modulus. If Poisson’s ratio for a material is known, along with just one of the three moduli, then the other two moduli for that material can be calculated, as shown by the mathematical relationships listed below in equations (4.19)–(4.21). Poisson’s ratio can take on values only within the range, 0 ≤ ≤ 0.5. Table 4.5 shows the extreme cases in which these values are reached. When stiffness and rigidity (E and G moduli) is very low compared to firmness (K modulus), then case A applies. On the other hand case B applies when a solid material has a low K modulus (low firmness, high compressibility) and very large stiffness and rigidity (large E and G). Table 4.6 shows the relationship between moduli for those cases.

4.2 Rheological Models

129

In reality, Poisson’s ratio lies between these extreme values for most materials. It can be calculated from [116]: E 3E · K 3K(1 − 2 ) = = 2(1 + ) 9K − E 2(1 + ) 

E·G 2(1 + ) E = =G K= 3(1 − 2 ) 9G − 3E 3(1 + 2 ) G=

E=

9G · K = 2G(1 + ) = 3K(1 − 2 ) 3K + G

E 3K − 2G E − 2G 1 − 3K = =

= 2G 2 2(3K + G)

(4.19) (4.20) (4.21)

(4.22)

4.2 Rheological Models Rheological models consisting of mechanical elements like springs and dashpots can be used to explain and interpret the rheological behavior of viscoelastic materials. The model is intended to behave qualitatively in a manner similar to that of the actual material. If the mechanical behavior can be expressed in terms of force–deformation or stress–strain and time, the results can lead to a mathematical equation. These equations can be used to explain, and in some cases predict the behavior of the material in response to various loading conditions. Thus, such models can be quite useful. Most rheological models are intended to show ideal behavior. The basic models most often used are those intended for ideal rigid, ideal viscous, ideal elastic and ideal plastic behavior. Figure 4.8 and Table 4.7 give an overview of the names and symbols used for these models. These basic mechanical elements can be further assembled into various combinations to produce more sophisticated mechanical models capable of

Figure 4.8. Symbols of mechanical elements used for basic rheological models. I Hookean element (spring), II breakage (rupture) element, III Newtonian element (dashpot), IV St.Venant element.

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4 Rheological Properties

Table 4.7. Basic rheological models name of model Pascal Newton St. Venant Hooke Euklid

rheological property ideal frictionless flow ideal viscous flow ideal plastic deformation ideal elastic deformation ideal inelastic (stiff, rigid)

application fluids fluids solids solids solids

simulating more complicated rheological behaviors, such as the models listed in Table 4.8. It is interesting to note that the Hookean spring and the Newtonian dashpot are the mechanical elements used most often in these rheological models. Note that they are the sole mechanical elements in the first two rheological models listed in Table 4.8. For example when used alone, the spring is a mechanical model representing ideal (Hookean) elastic behavior in solids. If we apply a force to the spring, it will extend a distance directly proportional to the applied force, with the constant of proportionality (modulus of elasticity) being represented by the spring constant. If we remove the applied force, the spring extension will fully recover. Likewise, the dashpot alone serves as a model for ideal viscous behavior exhibited by liquids. If we apply a constant force to the dashpot, it will extend continuously, representing flow, at a speed directly proportional to the applied force. A greater force will cause it to extend at a faster rate. The constant of proportionality between applied force and extension rate represents the viscosity of a liquid, and is called the viscosity of the viscous element (similar to spring constant for elasticity). Near the bottom of Table 4.8 are two more complicated models called the Maxwell model and the Kelvin model. Note that both these models consist only of spring and dashpot. In the Maxwell model both elements are combined in series, while in the Kelvin model they are combined in parallel. The Maxwell model is used to simulate stress relaxation in solid materials. This is a time-dependent rheological behavior characterized by the observation of stress decreasing over time in a sample subjected to a constant applied strain. The Kelvin model is used to simulate strain retardation (creep) in solid materials. This is also a time-dependent rheological behavior, but is characterized by the observation of strain increasing over time in a sample subjected to a constant applied stress. Application of the Kelvin and Maxwell models to creep and stress relaxation will be discussed more extensively in the section on texture measurement later in this chapter (see Section 4.6.1). The goal is to find the model which is best suited to describe the rheological behavior of the actual material of interest under conditions appropriate to the intended application. Examples of various such models are given in Table 4.8. Use of the Maxwell and Kelvin models in describing stress relaxation and creep behavior will be discussed later in Sections 4.6.1 and 4.6.2.

4.2 Rheological Models

131

Table 4.8. Combinations of mechanical elements to form basic rheological models element name

element symbol

Hooke

Newton

St.Venant

Pascal Euclid Maxwell

Prandtl

Kelvin

ideal flow without friction ideal inelastic (rigid)

same with breakage element

132

4 Rheological Properties Figure 4.9. Viscous behavior: In response to a shear stress, the deformation  continues to increase indefinitely, exhibiting the phenomenon of flow

4.3 Viscous Behavior { Flow Unlike solids, liquids cannot support their own weight, and are incapable of holding any shape. They must be contained in a vessel or surface depression. Otherwise, they will flow under the shear stress caused by their own weight. Therefore, the way in which liquids deform to an applied stress is to flow (continuous deformation). Recall when a shear stress causes a reversible angular deformation  on a solid, we call it an elastic shear deformation.When instead, a shear stress causes a continuously increasing strain  i.e. a constant ˙ , we call this flow. Flow is not elastic behavior, but is viscous behavior. Figure 4.9 appears the same as Figure 4.7 presented earlier to illustrate elastic shear on a solid. However, the same figure can also be used to illustrate ideal viscous behavior in a liquid. We only need to imagine that the upper surface with area A is a flat plate resting on top of a sample of liquid film having thickness y. When we pull on the flat plate with force F, we will be applying a shear stress to the liquid, and the liquid film will allow the plate to move in the direction of the force at a constant velocity dictated by the viscosity of the liquid. If we pull with greater force (increasing shear stress), the plate will move faster at a higher constant velocity. Therefore, not only do liquids respond to an applied stress by flowing, they respond to changes in stress by changes in the speed by which they flow (flow rate). Just as ideal elastic behavior of a solid material is a linear relationship between shear stress and shear strain governed by the shear modulus of elasticity G, ideal viscous behavior is a linear relationship between shear stress and the rate of shear strain, governed by the viscosity of the liquid. The rate of shear strain is the shear rate dy/dt. These relationships are expressed mathematically below. Ideal elastic behavior in solid:  =G·

(4.15)

Ideal viscous behavior in liquid:  = · ˙ where Ft A  G

tangential force in N area in m2 shear stress in N · m−2 shear modulus in N · m−2

(4.23)

4.3 Viscous Behavior – Flow

 ˙

133

angle of deformation in rad shear rate in s−1 viscosity in N · m−2 · s

Atoms and molecules in solids have stronger binding forces between them than in liquids or gases. Because of this, solids normally tend to show elastic behavior, whereas liquids and gases show viscous behavior. This is not a matter of molecular binding forces alone, but also of the competition between binding energy attracting molecules close together, and thermal energy driving molecules further apart. With rising temperature, the thermal energy can override the binding energy and a solid (elastic) changes into a liquid (viscous). On the other hand a solid can be brought to flow without raising the temperature by applying sufficiently high stress that the yield stress can be exceeded. For example, this can be done with a material like steel. Once the yield stress is exceeded, the steel will flow and the material will show permanent deformation. This is precisely what the blacksmith is doing when he strikes a hammer blow to a piece of steel. 4.3.1 Shear Rate It seems that one of the more difficult concepts to be grasped in the study of liquid rheology is a physical conceptualization of what is meant by rate of shear or shear rate. To help meet this objective, we will refer once again to Figure 4.9. Basically, the continuing change of the shear angle  deformation over time is called shear rate ˙ . The SI unit is s−1 . d (4.24) dt A more general way to look at the continuing shear angle deformation, is to view the deformation as the increasing angle caused the circular rotation of a radius on a fixed circle, as shown in Figure 4.10. The same deformation can be described by an angle  or by a distance along the perimeter ds: ˙ =

Figure 4.10. Angular deformation  and lateral (tangential) deformation ds

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4 Rheological Properties

Mathematically, these can be expressed as follows: ds r for small angles the approximation is valid tan  =

(4.25)

tan  = 

(4.26)

so  =

ds r

(4.27)

and ds d = dt dt · r Using the tangential velocity (see Figure 4.10)

(4.28)

ds (4.29) dt the shear rate is v (4.30) ˙ = r So the shear rate is the quotient of the tangential velocity and the radius of the rotational movement. So the shear rate is the same as the angular velocity of the rotation. Looking at Figure 4.9 again, the shear rate can be recognized as the quotient of the velocity of the upper plate and the distance to the (nonmoving) base plate, with: v=

tan  =

dx y

(4.31)

and with equation (4.26) the shear rate is ˙ =

dx vx = y · dt y

(4.32)

When then lower base plate in Figure 4.9 is no longer fixed, but allowed to move freely, then the more general definition of the shear rate has to be applied: ˙ =

dvx dy

(4.33)

˙ =

dv dr

(4.34)

vx vx2 − vx1 = y y2 − y1

(4.35)

or

with ˙ =

4.3 Viscous Behavior – Flow

135

Table 4.9. Terms used for the shear rate

Table 4.10. Terms for the shear angle

˙ /s−1 shear rate rate of shear shear velocity gradient rotational velocity angular velocity

 /rad shear angle shear shear deformation angular deformation

It is possible to calculate the shear rate between two moving plates. First of all, if the plates move together in the same direction at the same velocity, there is no relative difference in velocity, and the shear rate is zero. When the lower base plate is fixed (not moving), then equation (4.33) would apply: ˙ =

dvx vx v2 − v1 v2 − 0 v = ≈ = = dy y y2 − y1 y2 − 0 y

(4.36)

Imagine if in Figure 4.9 the upper plate were a round flat disk instead of a rectangular flat plate. Then a torque could be applied to induce a shear stress that would cause the circular plate to rotate upon the liquid film of thickness y, instead of moving (sliding) in a linear direction.Now,we can look at Figure 4.10 to help us recognize that the description of the shear rate in equations (4.33) and (4.36) are the same. In the case of the rectangular model in Figure 4.9, the finite element with would become a part of r. The SI unit in either case is s−1 . For example, from equation (4.33), we have , which algebraically reduces to s−1 . Example 4.2. Understanding shear rate during painting Using the symbols of Figure 4.9: (a) Assume a paint brush is moving at a velocity v = 0.5 m · s−1 with each stroke, and is leaving a paint film of 2 mm thickness. Then, we have: dv y = 2 mm, and the shear rate is ˙ = = 250 s−1 . dy (b) Assume a paint brush stroke velocity is changed to v = 1 m · s−1 , and is leaving a paint film of d = 0.2mm thickness. Then, we have: dv = 5000 s−1 . y = 0.2 mm, and the shear rate is ˙ = dy Example 4.3. Shear rate during roller drum process Using the symbols of Figure 4.10: (a) Assume a roller drum process produces a film layer with a thickness of y = 100 ‹m with a roll of radius r = 50 cm and 20 RPM. Then, the tangential velocity of the roll is: v = ! ·r =

20 · 2 · 0.5 m ≈ 1 m · s−1 60 s

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4 Rheological Properties

Table 4.11. Magnitudes and examples of shear rates process sedimentation of small solid particles sedimentation of larger solid particles flow caused by surface tension flow caused by gravity

shear rate ˙ /s−1 10−6 . . . 10−4

example

10−4 . . . 10−1

paint pigment, ceramic suspensions, spice particles in dressing dessert toppings, paint film, coatings,

10−2 . . . 10−1 10−1 . . . 101

fruit juice, medical suspensions

extrusion processing

100 . . . 103

roller drum processing pouring dipping chewing / swallowing flow through pipe stirring / mixing painting, brushing surface smearing spraying (aerosols) wet milling roller (pressing) homogenization (high pressure nozzle) lubrication calandering

101 . . . 102 101 . . . 102 101 . . . 102 101 . . . 102 100 . . . 103 101 . . . 103 102 . . . 104 102 . . . 104 103 . . . 106 103 . . . 105 104 . . . 106 105 . . . 106

pouring out of containers, flow of paint, dripping off excess coating snacks, cereals, tooth paste, noodles, pet food, polymers rolling of dough out of bottles, food, cosmetics dipping of confectionary in chocolate food, feed, pharmaceutical products pumping of liquds liquids paint, lip gloss, nail polish, spreads skin creams, salves, bath gels, hand lotion spray drying, spray painting, fuel injection grain processing, food suspensions, mustard printing of newspapers, lithography, fluid milk, soft-scrub scouring creams

103 . . . 107 103 . . . 107

motor oil, machine oil, axle grease polymer films, rubber sheeting

and the shear rate is: dv 1 m · s−1 = = 10000 s−1 ˙ = dy 10−4 m (b) Assume we change the conditions in the previous example to a thickness of y = 25 ‹m with a roll of radius r = 25 cm and 100 RPM. Then, the tangential velocity of the roll is 100 · 2 v =!·r = · 0.25 m = 2.6 m · s−1 60 s So, the shear rate is ˙ =

dv 2.6 m · s−1 = ≈ 105s−1 dy 25 · 10−6 m

4.3 Viscous Behavior – Flow

137

Example 4.4. Shear rate for liquid flow in a pipe or tube During laminar flow of an incompressible, ideal viscous fluid in pipe, the shear r·p 4·V˙ stress is given by W = 2l and the shear rate ˙W = ·r 2 . In a tube with a radius of 10 cm and at a flow rate of 6000 l/min, the shear rate between fluid and wall of the tube is: 4 · V˙ 4 · 6000 · 10−3m3 ˙W = = = 127 s−1 3  ·r 60 s ·  · (0.1 m)3 Example 4.5. Shear rate during sedimentation According to Stokes law, for Reynolds number < 1 and without interactions between particles and fluid, the terminal velocity of a particle falling through a fluid can be given by: v=

d2 · g (K − fl ). 18 ·

For sand particles with d = 0.5 ‹m in water, there is a difference between fluid and particle density of: K − fl = 1500 kg · m−3 − 1000 kg · m−3 = 500 kg · m−3 . With the dynamic viscosity of water of about = 1 mPa · s, there will be a terminal velocity of: (5 · 10−7 m)2 · 9.81 m · s−2 · 500 kg · m−3 = 6.8 · 10−8 m · s−1 18 · 10−3 N · m−2 · s So, the shear rate is: v=

˙ =

dv ∼ 6,8 · 10−8 m ∼ = = 0.1 s−1 dy 5 · 10−7 s · m

4.3.2 Newtonian Flow Behavior When there is a linear relationship between shear stress and resulting shear rate, the flow behavior is called ideal viscous or Newtonian. Fluids with that behavior are named Newtonian fluids. In the same sense, equation (4.23) is the classic mathematical equation that describes Newtonian flow behavior. It is the equation of a straight line passing through the origin.  = · ˙

(4.23)

In a diagram of shear stress versus shear rate the slope of this straight line gives the dynamic viscosity , which remains constant for all shear rates (see Figure 4.11). =

d d ˙

(4.37)

138

4 Rheological Properties Figure 4.11. Flow behavior curve (upper diagram) and viscosity curve (lower diagram) of a Newtonian fluid. The slope of the upper curve is constant for all shear rates.

or =

 ˙

(4.38)

Examples of common Newtonian fluids are listed in Table 4.12: Table 4.12. Newtonian fluids, examples material carbon dioxide nitrogen water ethanol milk olive oil glycerol

20 ◦ C /mPa · s 0.0148 0.0177 1.002 1.20 2 84 1490

Sometimes the kinematic viscosity  is used instead of the dynamic viscosity . The transformation simply is:  = (4.39) 

4.3 Viscous Behavior – Flow

where  

139

dynamic viscosity in Pa · s kinematic viscosity in m2 · s−1 density of fluid in kg · m−3 fluidity (flowability) in Pa−1 · s−1

The inverse quantity of the dynamic viscosity is called fluidity:

=

1

(4.40)

The SI units for dynamic viscosity are N · m−2 · s, which reduces to Pa · s. Water at room temperature has a dynamic viscosity of about = 1 mPa · s. With a density of about 1000 kg·m−3,the kinematic viscosity is about  = 10−6 m2 ·s−1 . For conversion, see Table 4.13. Table 4.13. Conversion of older viscosity units dynamic viscosity

kinematic viscosity

Poise (P)

Stokes (St)

Poiseuille (Pl) 1 P = 1 dyn · s · cm−2 =

10−5 N · s = 10−1 Pa · s 10−4 m2

1 St = 1 cm2 · s−1 = 10−4 m2 · s−1 1 St = 10−6 m2 · s−1

1 cP = 10−2 P = 10−3 Pa · s = 1mPa · s 1Pa · s = 1 Pl value of water = 1 cP

value of water = 1 cSt

In the case of Newtonian fluids the slope of the straight line on the shear stress versus shear rate diagram is constant (see Figure 4.11). This means the dynamic viscosity, which is the derivative of shear stress with respect to shear rate, = dd˙ must also be a constant, and will be the same for all shear rates. So, in the case of Newtonian fluids, a single value for is sufficient to describe the flow behavior for all shear rates. It is a “universal” viscosity for all situations of laminar flow in which the material is used. This is not the case for most biological materials and foods: For many of these materials, the viscosity is not the same at different shear rates. We call this non-Newtonian behavior. 4.3.3 Non-Newtonian Flow Behavior If there is no linear relationship between shear stress  and shear rate ˙ or if the flow behavior curve does not pass through the origin (0,0) on the shear

140

4 Rheological Properties Figure 4.12. Flow behavior curves. 1: Newtonian, 2: pseudoplastic (shear thinning), 3: dilatant (shear thickening), 4: Bingham plastic (linear with yield stress), 5: Herschel–Bulkley or mixed plastic (pseudoplastic with yield stress)

Figure 4.13. Viscosity curves 1: Newtonian, 2: pseudoplastic, 3: dilatant, 4: Bingham plastic, 5: Herschel–Bulkley

stress–shear rate diagram, we call this non-Newtonian flow behavior. The flow behavior of non-Newtonian fluids depends on the stress conditions to which the fluids are subjected. Some typical flow behavior curves are summarized in Figure 4.12. 4.3.4 Comparison of Newtonian with Non-Newtonian Fluids Newtonian fluids have the same viscosity at low shear rates as at high shear rates. In contrast, non-Newtonian fluids do not have a constant viscosity with respect to shear rate. Their viscosity will depend on stress conditions (shear rate),and often also on time.Figure 4.14 is a viscosity versus shear rate diagram on which viscosity curves for two different non-Newtonian fluids are shown, along with the horizontal straight line representing the constant viscosity of a Newtonian fluid.At low values of shear rate ˙1 , the viscosity of fluid 3 is higher than that of fluid 2. However, as shear rate increases to a higher value ˙2, now the viscosity of fluid 3 is lower than that of fluid 2. The viscosity of fluid 1 is the same over all shear rates. This example shows that for non-Newtonian fluids, it is not sufficient to provide one single value of viscosity to describe the flow behavior. Instead, it

4.3 Viscous Behavior – Flow

141 Figure 4.14. Newtonian fluid (1), and non-Newtonian fluids (2,3) in comparison

is necessary to say “viscosity at shear rate zzz.” It would be far more useful, however, to provide the total flow behavior curve or viscosity curve. There are ways to specify a total flow behavior curve with numerical values for the constant parameters in the mathematical equations describing the flow behavior curves (see later in the chapter). For a very small interval in the range of shear rates of interest, a nonlinear flow curve may be approximated by a linear function. That means a nonNewtonian fluid can be treated approximately as a Newtonian fluid within a small interval of shear rates. Whether an approximation is acceptable depends on many factors like the type of material, the nature of question being asked, and the intended use of the result. 4.3.5 Pseudoplastic Flow Behavior Pseudoplastic flow behavior occurs when we observe shear stress increasing at a diminishing rate with increasing shear rate. On a shear stress–shear rate diagram, the flow behavior curve has a convex profile in which the tangential slope is decreasing with increasing shear rate. This means the viscosity is decreasing with increasing shear rate. Sometimes this observation is referred to as “shear-thinning” behavior. Figure 4.12 and Figure 4.13 show examples of the flow behavior curve and viscosity curve for a pseudoplastic fluid. It is characteristic of pseudoplastic fluids that the increasing flow resistance (shear stress) seems to decrease when the fluid is subjected to higher shear rates. This behavior is assumed to be caused by decreasing molecular interactions within the molecular structureof the fluid during flow. In the case of flowing macromolecules, the additional effects of unfolding and reorientation of molecules may be occurring. When these effects are completely reversible, we call this “true” pseudoplasticity. This means that if we first increase shear rate to a given value, and then decrease it back again, the viscosity will be fully restored to the same value as before. Sometimes the effect of shear-thinning is not completely reversible. This can be explained by a permanent loss of structure in the material under inves-

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4 Rheological Properties

Table 4.14. True and apparent pseudoplasticity pseudoplasticity behavior example

true reversible aqueous xanthan solution

apparent irreversible yogurt

tigation (e.g. yogurt shows this effect). In that case we have no “true” pseudoplasticity, but an apparent pseudoplasticity. Table 4.14 shows some examples. 4.3.6 Thixotropic Flow Behavior Thixotropic flow behavior is observed when shear stress or viscosity decreases over time at a constant shear rate. The reason for the decreasing viscosity (or shear stress) is assumed to be caused by a decrease in the intermolecular interactions within the molecular structure of the material. When shear rate stops, the original structure is restored, as well as the initial viscosity. This is called “true” thixotropicity. If the viscosity is not restored because of irreversible structure breakdown, the behavior is called apparent thixotropicity. Many of us have experienced thixotropicity when stirring paint. When first beginning to stir the paint, a great effort is needed (shear stress) to get the paint moving at a given rate of stirring. But as time goes on, it seems that less and less effort is needed to maintain the rate of stirring, because the viscosity is decreasing over time at constant shear rate. This is thixotropicity. 4.3.7 Dilatant Flow Behavior Dilatant flow behavior occurs when we observe shear stress increasing at an increasing rate with increasing shear rate. On a shear stress–shear rate diagram, the flow behavior curve has a concave profile in which the tangential slope is increasing with increasing shear rate. This means the viscosity is also increasing with increasing shear rate. Sometimes this observation is referred to as “shear-thickening” behavior. This flow behavior is seen in highly concentrated suspensions. Figure 4.12 and Figure 4.13 show examples of the flow behavior curve and viscosity curve for a dilatant fluid. It is assumed that under increasing shear rate the liquid between the solid particles is squeezed out and the friction between the particles increases. This increasing friction accounts for the increasing shear stress and increasing viscosity experienced as shear rate increases. Also, the squeezing causes a dilatation (an increase in volume) of the suspension, and therefore the phenomenon is called dilatant flow behavior. When these effects are completely reversible, we call this “true” dilatancy. This means that if we first increase shear rate to a given value, and then decrease it back again, the viscosity will be fully restored to the same value as before. Sometimes the effect of shear-thickening is not completely reversible. This can be explained by a permanent damage to the

4.3 Viscous Behavior – Flow

143

molecular structure in the material under investigation. In that case we have no “true” dilatancy, but an apparent dilatancy. 4.3.8 Rheopectic Flow Behavior Rheopectic flow behavior is observed when shear stress or viscosity increases over time at a constant shear rate. The reason for the increasing viscosity (or shear stress) is also assumed to stem from the intermolecular interactions causing friction to increase with time at constant shear rate within the molecular structure of the material. When shear rate stops, the original structure is restored, as well as the initial viscosity. This is called “true” rheopecticity. If the viscosity is not restored because of irreversible structure damage, the behavior is called apparent rheopecticity. When observing what appears to be rheopectic behavior, it is important that we interpret correctly what we observe. For example, when whipping liquid heavy cream into whipped cream, we observe the need for more and more effort to maintain the same whipping speed (viscosity increasing over time at constant shear rate). This would appear to be rheopectic behavior. But, that is not the case here. In whipping cream, we start out with a liquid of relatively low viscosity that becomes transformed into a soft-solid foam as a result of incorporating air into the cream producing a gas–liquid emulsion.We now have a totally new material with different rheological properties, and not a liquid with rheopectic behavior. 4.3.9 Plastic Flow Behavior When a solid material keeps its deformation as a permanent set after taking away the shear stress,we call that plastic deformation.Liquids will demonstrate plastic behavior when they will not begin to flow until a minimum shear stress is exceeded that allows them to yield and begin to flow. This initial stress that must be overcome before the liquid will yield to begin flow is called the yield stress.So,the yield stress is the minimum shear stress needed to get the material flowing, and it is a characteristic of plastic flow behavior. An example is steel which can be deformed if the shear stress from a hammer blow exceeds this value.Another example is butter, which looks like a solid, but can be deformed, and keeps its deformation as a permanent set. When a solid body is loaded with a shear stress below its yield stress, it can be deformed elastically, but will not flow. In a flow behavior curve on a shear stress–shear rate diagram, the yield stress can be easily found as the intercept on the stress axis at zero shear rate (see Figure 4.12). Because the viscosity (Figure 4.13) mathematically is the derivative of the flow behavior curve, the yield stress on the viscosity curve appears to originate as an indefinitely high viscosity. This can be interpreted as a nonflowing solid, which must have infinite viscosity.

144

4 Rheological Properties

Reasons for plasticity in liquids also stem from strong intermolecular interactions between the molecules within the molecular structure of the material. The stronger are these interactions, the higher the yield stress. As with many other properties, the yield stress also depends on temperature. For example butter kept cold in the refrigerator will have a much higher yield stress than if the butter were left out at room temperature. This is the reason why the blacksmith prefers to work with hot steel in order to get more yield (deformation) with each hammer blow. In the preparation of certain foods, it is important to achieve a needed yield stress. For example, whipped cream must have a sufficiently high yield stress in order to hold its shape as a decorative dessert topping. If the yield stress is too low, it will begin to flow and lose its shape, and we cannot use it for a dessert topping. From a scientifically pure standpoint, all materials have a yield stress, but it is sometimes so low that we do not notice it,such as with water or air.Therefore, for practical reasons, we define plastic materials as those with a measurable (observable) yield stress. Normally this is defined as having a yield stress of such magnitude that they do not flow by themselves, meaning they will not flow under the force of their own weight in response to gravity. Example 4.6. An ice cream bar shall be dipped into molten chocolate to get a chocolate coating of 2 mm thickness. To what temperature should the molten chocolate be heated in order that it will leave this coating thickness? Assume density of chocolate is 1235 kg · m−3 . Temp. in ◦ C 37.5 38.0 38.5 39.0 39.5

yield stress in Pa 32 28 25 22 19

Figure 4.15. Chocolate coating with thickness d0

Looking at Figure 4.15 the weight force of the chocolate layer with the thickness d is FG = m · g =  · A · d · g The final coating thickness will be left on the bar when the weight force of the coating acting downward is equal to the shear force caused by the yield stress acting upward. So, the shear stress caused by this layer is set equal to weight stress:

4.3 Viscous Behavior – Flow

145

FG m · g  · A · d · g = = =  ·d ·g A A A To have the desired thickness d0, the excess molten chocolate must stop flowing at yield stress 0 =

0 =  · d0 · g 0 = 1235 kg · m−3 · 2 · 10−3 m · 9.81 m · s−2 = 24 Pa Using the data given, we get a temperature of 38.4 ◦ C, which we should choose for the molten chocolate to achieve this 2 mm coating. Here 0 yield stress in Pa A area in m2 d thickness in m  density in kg · m−3 g gravitational acceleration in m · s−2 FG weight force in N m mass in kg 4.3.10 Overview: Non-Newtonian Flow Behavior In all cases of non-newtonian flow, the flow behavior curve on a shear stress– shear rate diagram is not a straight line through the origin (see Figure 4.11). Because only a Newtonian fluid has a viscosity which is constant over time, we call those cases of non-Newtonian fluids, in which the flow properties (especially viscosity and yield stress) are dependent on time, time-dependent behaviors. These behaviors are clearly evident in thixotropicity and rheopecticity. Figure 4.16 shows a schematic of non-Newtonian flow behavior.

Figure 4.16. Non-Newtonian flow behavior, schematic of behavior categories and their terminology. Examples are: A: paint, tomato paste, B: crystallized honey, C: whipped eggwhite, solid butter, lipstick, D: wheat dough, E: ketchup

Materials which show plastic flow are also non-Newtonian materials, even when the viscosity is not dependent on shear stress and time. Flow behavior like this is called Bingham plastic flow, and the material is called a Bingham plastic fluid (see Figure 4.11).

146

4 Rheological Properties

Table 4.15. Glossary of flow behavior terms Newtonian non-Newtonian pseudoplastic dilatant plastic thixotropic rheopectic

the flow curve is a straight line through the origin the flow curve is not a straight line through the origin viscosity decreases with increasing shear rate (shear thinning) viscosity increases with increasing shear rate (shear thickening) the fluid has a yield stress viscosity decreases over time at constant shear rate viscosity increases over time at constant shear rate

Real materials normally have a more complicated flow behavior which can be explained with various mixtures of the different ideal behaviors described up to here. Sometimes approximations can help, like treating the Bingham plastic fluid as a Newtonian fluid once the initial yield stress has been discounted. For many food science and engineering applications, it is convenient to transform complicated flow behavior curves and viscosity curves into mathematical equations with constant parameters. Then only a few numbers (parameters) are sufficient to represent and to specify the complete curves. For this purpose, model functions have to be derived, and tested and challenged to be sure of their validity. 4.3.11 Model Functions Rheological model functions are nothing more than the mathematical equations derived to describe the various flow behavior curves on shear stress–shear rate diagrams. First we will focus on model functions for fluids without yield stress. Then, we will address fluids with yield stress. Pseudoplastic fluids are assumed to have an initial viscosity at the origin (˙ = 0) = 0 which then decreases with increasing shear rate. The decrease is assumed to be the consequence of (reversible) loss of structure/network in the material (in the German language pseudoplastic is called strukturviskos because of this interpretation). When the molecular structure/network in the fluid has reached a steady state (where the intermolecular forces acting to build the structure/network and those acting to break it down by shear are in equilibrium) no further decrease of viscosity is observed. This viscosity is called equilibrium viscosity (˙ = ∞) = ∞ . Figure 4.17 shows a typical profile for a flow curve of this type. The viscosity curve (which is the derivative of the flow behavior curve) can also be constructed, and is given in Figure 4.18. In this type of pseudoplastic material, the viscosity is changing from 0 to ∞ with increasing shear rate. For a mathematical description of these sigmoid shaped curves, a number of model functions have been developed. In general, most of these functions require two or three parameters. These parameters are constants in the model function, whose numerical values make

4.3 Viscous Behavior – Flow

147 Figure 4.17. Flow behavior curve during structure loss in a pseudoplastic fluid (schematic)

Figure 4.18. Viscosity curve during structure loss in a pseudoplastic fluid (schematic)

the function specific for a given material.Therefore,in order to use these model functions,these parameters have to be known.Normally,model parameters are measured experimentally, or they be obtained from other investigators working on the same material, who report their findings in the scientific literature. Table 4.16 lists some model functions like these. There are cases where a model function takes on the same form as that of a simpler more easily recognized model function, like Newton’s law. In other words, these model functions can be thought as more or less sophisticated modifications of basic functions like which describes Newtonian flow behavior. A classic question posed by investigators working on rheology of fluids is, “Which model is the right one for my material and/ or application?” The answer will depend on factors like: for what purpose is the viscosity needed? In what range of shear rates will the work be done? How precise must the data be? These models differ in their ability for preciseness or “goodness of fit” in approximating experimental data. For example, the model functions after Ferry, Steiner–Steiger–Ory, De Haven, Ostwald–de-Waele, Ellis I, and Sisko are valid only in the region < ∞ where the fluid structure is not completely broken down.

148

4 Rheological Properties

Table 4.16. Model functions for fluids without yield stress, examples name Newton

model function for flow behavior curve  = · ˙

model function for viscosity curve  = ˙

Ferry

=

0 · ˙ 1